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\usepackage[def-file=diffcoeff5,spaced=-1]{diffcoeff}

% nabla in numer, d in denom
\difdef { f, s } { n } 
  {
    op-symbol     = \nabla,
    op-symbol-alt = \mathrm{d}
  }
% no sub nudge (a sea of white space)
\difdef { fp } { wsp } 
  { sub-nudge = 0 mu }
% align op left; no italic nudge 
\difdef { f } { left0 } 
  {
    *op-set-left  = true,
    *italic-nudge = 0 mu
  }
% align op left; italic nudge
\difdef { fp } { left } 
  {
    op-symbol      = \partial,
    op-order-nudge = 1 mu,
    *op-set-left  = true,
    *italic-nudge = 3 mu
  }
% partial variant of \diff
\difdef { f } { p }
  {
    op-symbol      = \partial,
    op-order-nudge = 1 mu
  }
% partial, 3mu sep of terms
\difdef { fp, sp } { 3mu } 
  {  multi-term-sep  = 3 mu }
% D jacobian
\difdef { j } { D } 
  { op-symbol = D }
% square bracket jabobian
\difdef { j } { [ } 
  { 
    lvwrap-Ldelim = \onemu\mleft [,
    lvwrap-Rdelim = \mright ] 
  }
% mimicking the \Braket command
% of the braket package
\difdef{ s }{ bk }
  { 
      slash-tok = ,
      op-symbol =  ,
      multi-term-sep = 3mu\middle|\mskip3mu ,
      outer-Ldelim=\left\langle ,
      outer-Rdelim=\right\rangle
  }
\NewDocumentCommand \Braket { m }
  { 
    \difoverride {\negmu}
  	\difs.bk.{}{#1}[] 
  }

\ifdefined\showcaptionsetup
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 \PassOptionsToPackage{caption=false}{subfig}
\fi
\usepackage{subfig}
\makeatother

\begin{document}
\title{\texttt{diffcoeff}~\\
 a \LaTeX{} package to ease\texttt{ }the~\\
 writing of differential coefficients \\
 Version 5.1}
\author{Andrew Parsloe\\
{\small (ajparsloe@gmail.com)}}
\maketitle
\begin{abstract}
\noindent \verb`diffcoeff` is a \LaTeX{} package to ease the writing
of ordinary, partial and other derivatives of arbitrary algebraic
or numeric order. For mixed partial derivatives, the total order of
differentiation is calculated by the package. Optional arguments allow
for points of evaluation (ordinary derivatives), or variables held
constant (partial derivatives), and the placement of the differentiand
in numerator or appended. Besides $\diff yx$, forms like $\difs yx$
and $\difcp yx$ are also available, as well as derivatives built
from $D$, $\Delta$, $\delta$, and configurable jacobians and differentials.
Other notations like line elements ($\dl.+.{x,y,z}^{2}$) and bra-kets
are easily produced. 

Version 5 of \verb`diffcoeff` more fully embraces the configurability
offered by \verb`xtemplate` than earlier versions. Some incompatibilities
have arisen, but all is not lost: version 4 is still available with
the command

\verb`\usepackage[<options>]{diffcoeff}[=v4]`
\end{abstract}
\noindent\begin{minipage}[t]{1\columnwidth}%
\begin{shaded}%

\subsubsection*{For users of version 4}

The \verb`\diff` and \verb`\diffp` commands of version 4 of \verb`diffcoeff`
remain, but lack the `spacing switch' \verb`!` (which on reflection
was a mistake) and, more noticeably, the `slash switch' \verb`/`.
Slash-fraction derivatives are now created with the \verb`\difs`
and \verb`\difsp` commands. New commands \verb`\difc` and \verb`\difcp`
produce derivatives in `compact notation' like $\difc yx$ and $\difcp yx$.
To avoid cluttering formulas with a second square-bracket delimited
optional argument before the differentiand, the order-override option
has been replaced by a new command \verb`\difoverride`. The order
of differentiand and variable(s) of differentiation can now be reversed,
when the differentiand is appended, by using a second star, e.g.,
\verb`\diffp**`. The two-argument \verb`\diffdef` command of earlier
versions has been replaced by the three-argument command \verb`\difdef`,
the additional argument determining which one or more of the \verb`f`,
\verb`s`, \verb`c`, \verb`fp`, \verb`sp` or \verb`cp` forms the
defined variant applies to. The differential command \verb`\dl` has
been rewritten and is now fully template-configurable (allowing easy
writing of line elements like $\dl.+.{x,y,z}^{2}$), and the jacobian
command \verb`\jacob` is also configurable. Indeed version 5 of \verb`diffcoeff`
more fully embraces the configurability offered by the \verb`xtemplate`
package than previous versions, bringing other notations -- like
those of the \verb`braket` package -- within its compass.

\subsubsection*{ISO defaults}

The \verb`ISO` package option is redundant. Unlike the default set-up
in version 4, the defaults in version 5 of \verb`diffcoeff` are chosen
to reflect ISO recommendations (see the standard ISO 80000-2). In
particular this means upright `d's and subscripted parentheses enclosing
a derivative to indicate a point of evaluation. This document is written
with those defaults. For those (like the author) who prefer math-italic
`d's and a subscripted vertical rule to indicate a point of evaluation,
the means of creating `variant forms' or changing the defaults is
readily available; see §§\ref{sec:Variant-forms}, \ref{sec:ChangingDefaults}. \end{shaded}%
\end{minipage}

\tableofcontents{}

\chapter{Introduction}

\verb`diffcoeff.sty` is written in the \verb`expl3` language of
\LaTeX 3, now part of standard \LaTeX{} since February 2020. A \LaTeX{}
distribution from or later than that date is assumed. The package
also requires the packages \verb`xtemplate` (part of the \verb`l3packages`
bundle) and \verb`mleftright`. The package is loaded in the usual
way by entering 
\begin{verbatim}
    \usepackage{diffcoeff}
\end{verbatim}
in the preamble of your document or, if package options are being
used,\footnote{Angle brackets indicate possible user input (\emph{without} the angle
brackets).}
\begin{verbatim}
    \usepackage[<options>]{diffcoeff} 
\end{verbatim}
The interface of \verb`diffcoeff` with version 5 has changed from
earlier versions. If you want the familiar behaviour of version 4,
add to these commands a trailing optional argument like this,
\begin{verbatim}
    \usepackage[<options>]{diffcoeff}[=v4]
\end{verbatim}
(with no space after the `=' sign!). Of course you will not get
the new features of version 5. Working with version 4 is described
in the document \verb`diffcoeff4.pdf`.

\section{Package options}

\label{sec:Package-options}There are four package options, which
are entered in a comma-separated list in the optional argument of
the \verb`\usepackage` command. The \emph{second} and \emph{fourth}
are new with version 5 of \verb`diffcoeff`.
\begin{enumerate}
\item The first package option is the \verb`spaced` option which takes
three values:
\begin{enumerate}
\item \verb`spaced=1` inserts a small space before the differentiand; this
is the default so that entering \verb`spaced` is equivalent to \verb`spaced=1`;
\item \verb`spaced=0` inserts no space before the differentiand; \verb`diffcoeff`
is initialized to \verb`spaced=0` so that if the spaced option is
not used \verb`spaced=0` is assumed;
\item \verb`spaced=-1` inserts a small space before the differentiand \emph{if
it contains more than one token}, and no space otherwise. The present
document uses \verb`spaced=-1`.
\end{enumerate}
This option is discussed in §\ref{subsec:Spacing-before-derivand}.
\item For the second package option, by entering \verb`mleftright` in the
options list the command \verb`\mleftright` is automatically inserted
in the preamble. The effect is to change all occurrences of \verb`\left`,
\verb`\right` in the document to \verb`\mleft`, \verb`\mright`
so that the spacing around scalable delimiters modified by \verb`\left`,
\verb`\right` is reduced; see the box below and §\ref{subsec:mleftright}.
The present document does not use this package option. 
\item The third package option requires the \verb`<filename>` of a file
with extension \verb`.def`, \verb`<filename>.def`, containing definitions
of variant forms (see §\ref{sec:Variant-forms}) of derivative:
\begin{verbatim}
    \usepackage[def-file=<filename>]{diffcoeff}
\end{verbatim}
This is discussed in §\ref{subsec:The-.def-file}. The present document
uses the package option \verb`def-file=diffcoeff5`.
\item The fourth package option \verb`DIF` is a comma list of \emph{key=value}
statements amending the built-in defaults for the `grandparent'
template \verb`DIF`; see §\ref{subsec:Changing-DIF-defaults}. The
present document does not use this package option.
\end{enumerate}
\noindent\begin{minipage}[t]{1\columnwidth}%
\begin{shaded}%
To see the effect of the \verb`mleftright` package, consider the
expression
\begin{centred}
\verb`\[ \ln \left(\frac xy\right),\quad\sin\left(x^2\right). \]`
$\Longrightarrow$ \[ \ln \left(\frac xy\right),\quad\sin\left(x^2\right). \]
\end{centred}
in which there is significant whitespace before and after the parentheses.
The package \verb`mleftright` enables this whitespace to be reduced
by using \verb`\mleft`, \verb`\mright` in place of \verb`\left`,
\verb`\right`:
\begin{centred}
\verb`\[ \ln \mleft(\frac xy\mright),\quad\sin\mleft(x^2\mright). \]`
$\Longrightarrow$ \[ \ln \mleft(\frac xy\mright),\quad\sin\mleft(x^2\mright). \]
\end{centred}
\end{shaded}%
\end{minipage}

\noindent %
\noindent\begin{minipage}[t]{1\columnwidth}%
\begin{shaded}%
If you put \verb`\mleftright` in the preamble, which is what the
\verb`mleftright` package option does, then all occurrences of \verb`\left`,
\verb`\right` in the document will be affected. \verb`\left`, \verb`\right`
can be restored to their normal behaviour by the command \verb`\mleftrightrestore`.
Rather than use \verb`\mleft`, \verb`\mright` explicitly, as in
the example, the same effect can be obtained by using \verb`\left`,
\verb`\right` and preceding the expression with the command \verb`\mleftright`:
\begin{verbatim}
  \mleftright 
  \[ \ln \left(\frac xy\right),\quad\sin \left(x^2\right). \]
  \mleftrightrestore
\end{verbatim}
$\Longrightarrow$   \mleftright 
  \[ \ln \left(\frac xy\right),\quad\sin \left(x^2\right). \]
  \mleftrightrestore \end{shaded}%
\end{minipage}\medskip{}
For the present document, the call is
\begin{verbatim}
    \usepackage[def-file=diffcoeff5,spaced=-1]{diffcoeff}
\end{verbatim}

\section{A Rogues' Gallery of derivatives}

\label{sec:Rogues'-gallery}Browsing through some (rather old) calculus
textbooks and texts on statistical mechanics, relativity and classical
mechanics I find the following choice examples of derivatives `disporting
every which way'.
\begin{itemize}
\item Multi-character variables of differentiation un-parenthesized:
\begin{equation}
\diffp{\frac{\psi}{\Theta}}{\frac{1}{\Theta}},\quad\diffp{E/T}{1/T},\quad\diff{\ln f}{\ln x_{0}},\quad\diffp\psi{a_{i},\frac{1}{\Theta}},\quad\diffp{\mathcal{L}}{\eta_{,i}^{(r)}}.\label{eq:eg1}
\end{equation}
\item Multi-character variables of differentiation parenthesized in \emph{higher-order}
derivatives, where the parentheses do not or (sometimes) do include
the operator: 
\begin{equation}
\diffp[2]q{\frac{1}{\Theta}},\quad\diffp[2]q{1/\Theta},\quad\diffp[2]\varepsilon{a_{i}},\quad\diff.(dv).[2]{\phi^{i}(x^{i})}{x^{i}}.\label{eq:eg3}
\end{equation}
Should the $\mathrm{d}$ or $\partial$ be included within the parentheses,
as in the last of (\ref{eq:eg3}), or not, as in the others? Logic
says `yes'; practice suggests (generally) `no'.
\item Indicating a point of evaluation is similarly varied: 
\begin{equation}
\diffp.|.\phi\varepsilon[\varepsilon=\varepsilon_{0}],\quad\diff.|.[2]\phi\varepsilon[\varepsilon=\varepsilon_{0}],\quad\diffp.].{b^{\beta}}{a^{\alpha}}[b=0],\quad\diff uv[v=0].\label{eq:eg4}
\end{equation}
ISO 80000-2 (item 2.11.13) favours the last of these -- parentheses
-- for ordinary derivatives. Presumably, partial derivatives should
follow suit, although parentheses are also used to indicate variables
held constant: 
\begin{equation}
\diffp*{\frac{P}{T}}U[V],\quad\diffp S{N_{2}}[U,V,N_{1}],\quad\difsp ST[V]\,.\label{eq:eg5}
\end{equation}
\item Other symbols besides $\mathrm{d}$ and $\partial$ are used to denote
derivative-like quantities. From introductory calculus and from classical
mechanics and thermodynamics come $\delta$ and $\Delta$, from fluid
mechanics comes $\mathrm{D}$: 
\begin{equation}
\diff.gd.yx,\quad\diff.D'.\rho t,\quad\diff.gD.UT[V],\quad\difs.gD.UT,\quad\diff.gd.{\mathcal{L}}{\eta^{(r)}}.\label{eq:eg6}
\end{equation}
\item There are those, like the International Organization for Standardization
(ISO), who stipulate (or prefer) an upright $\mathrm{d}$ for their
derivatives, and there are those (like the author, through sixty years
of habit) who prefer a math-italic $d$: 
\begin{equation}
\diff yx,\qquad\diff.d'.yx,\label{eq:eg11}
\end{equation}
and of course also in slash-fraction form $\difs yx$, $\difs.d'.yx$.
Subscripted forms of derivative are also used -- for example, $\difcp Fx$,
or in the discussion of differential equations one sometimes comes
across expressions like 
\[
\difc.bD.[2]y{x}+2\difc.bD.y{x}-4=0.
\]
\item When the differentiand is too big or awkward to sit in the numerator
and is appended to the operator, the $\mathrm{d}$ or $\partial$
in the numerator is generally centred -- but not always. In texts
prior to the age of computerised typesetting one will sometimes find
the symbol pushed to the \emph{left}: 
\begin{equation}
\diffp.left.*{\diffp.left.{x^{i^{*}}}{x^{k^{*}}}}{x^{l^{*}}},\quad\diff.left0.*{\left(\frac{m\mathbf{q}_{x}}{\sqrt{1-q^{2}}}\right)}{t}.\label{eq:eg12}
\end{equation}
The keen-eyed will note an italic adjustment with the first expression,
so that the $\partial$s in the numerators are indented a little (to
line up -- more or less -- in a slanting column with the $\partial$s
in the denominators).
\item Then there is the case when the operator in the numerator differs
from that in the denominator. For instance, in tensor calculus acceleration
is sometimes written 
\[
\diff.n.{v^{i}}t=\diff{v^{i}}t+\Gamma_{k\hphantom{i}h}^{\hphantom{k}i}v^{h}\diff{y^{k}}t
\]
where $\nabla v^{i}$ is the `absolute differential' of the velocity
$v^{i}$.
\end{itemize}
The \verb`diffcoeff` package has the generative power to cope with
all these variations -- see §\ref{sec:Variant-forms} -- although
it is unlikely an author should need to call on this capacity to anything
like the extent required for this Rogues' Gallery.

\chapter{Syntax and use}

\verb`diffcoeff` aims to ease the writing of derivatives (sometimes
also called differential coefficients). There are long-established
shorthands available in a few cases: $\dot{x}$ and $\ddot{x}$ for
the time derivatives of a function $x$ of time $t$; $y'$ and $y''$
for the derivatives of a function $y$ (usually) of $x$. But mostly
derivatives are expressed in fraction form and require more keystrokes
to compose. It is here that \verb`diffcoeff` is aimed. It uses three
pairs of commands: \verb`\diff` and \verb`\diffp` to write (upright)
fraction forms of ordinary and partial derivatives like $\diff yx$,
$\diffp yx$, generally intended for display-style environments; \verb`\difs`
and \verb`\difsp` for slash-fraction forms of ordinary and partial
derivatives like $\difs yx$, $\difsp yx$, generally intended for
text-style environments; and \verb`\difc` and \verb`\difcp` to write
compact forms of ordinary and partial derivatives like $\difc yx$
and $\difcp yx$. (Of these, the `s' forms replace the slash argument
for the \verb`\diff`, \verb`\diffp` commands in version 4 of \verb`diffcoeff`,
and the `c' form is new to version 5.\cprotect\footnote{Suggested by a question on \TeX{} StackExchange: \url{https://tex.stackexchange.com/questions/652223/write-a-derivative-operator-without-denominator-using-diffcoef/652298#652298}})

\subsubsection*{Note}

I refer throughout to the quantity or function being differentiated
as the \emph{differentiand} or \emph{derivand }(in line with \emph{integrand},
\emph{operand}, etc.) and shall sometimes use \verb`\difx` (resp.
\verb`\difxp`) to make general statements about any or all of \verb`\diff`,
\verb`\difs` or \verb`\difc` (resp.\verb`\diffp`, \verb`\difsp`,
\verb`\difcp`).

\section{Syntax}

All commands, \verb`\difx`, \verb`\difxp`, share the same syntax.
With options present the syntax is
\begin{verbatim}
    \difx.name.*[order-spec]{differentiand}
        {variable(s)}[pt of eval]

    \difx.name.**[order-spec]{variable(s)}
        {differentiand}[pt of eval]
\end{verbatim}
The syntax is identical for \verb`\difxp`. The seven arguments have
the following meanings:
\begin{itemize}
\item \verb`name` (optional) A dot-delimited name to distinguish a variant
form (non-default form) of derivative; not discussed further until
§\ref{chap:Templates-defaults-variants} below, and specifically,
§\ref{sec:Variant-forms}.
\item \verb`*` (optional) The presence of a star (asterisk) signals: \emph{append}
the differentiand; its absence means the differentiand appears in
the numerator of an upright- or slash-fraction form derivative; no
effect for compact-form derivatives unless (see next) a second \verb`*`
is present; see §\ref{subsec:Appending-the-differentiand}.
\item \verb`*` (optional) The presence of a \emph{second} star signals
that the argument specifying the variable(s) of differentiation comes
\emph{before} the argument specifying the differentiand; this is sometimes
convenient when a complicated or lengthy differentiand is appended;
see §\ref{subsec:Transposing-the-argument}.
\item \verb`order-spec` (optional) The order of differentiation when differentiating
in a single variable, or a comma list of orders of differentiation
for a mixed partial derivative; see §\ref{subsec:Higher-order-derivatives}
and §\ref{subsec:Mixed-partial-derivatives}.
\item \verb`differentiand` (mandatory) The function being differentiated.
\item \verb`variable(s)` (mandatory) The variable of differentiation or
a comma list of variables of differentiation (for a mixed partial
derivative).
\item \verb`pt of eval` (optional) Point of evaluation or, for partial
derivatives, variable or variables held constant; \emph{no space}
before the left square bracket; see §\ref{subsec:Point-of-evaluation}.
\end{itemize}

\section{General use}

\label{sec:General-use}Writing \verb`\diff{y}{x}` will produce $\diff{y}{x}$
in an inline math environment (i.e., placed between \verb`\( \)`
or \verb`$ $`) or 
\[
\diff{y}{x}
\]
in display style (placed, for instance, between {\ttfamily\verb`\[ \]`}
). In fact \verb`\diff yx` (omitting the braces) will produce these
results, with a saving on keystrokes. The braces are needed only when
an argument -- the variable of differentiation, or the differentiand
-- is multi-token:
\begin{centred}
\verb`\[ \diff{\ln x}x \]` $\Longrightarrow$ \[ \diff{\ln x}x\]
\end{centred}
\begin{itemize}
\item If you want math-italic `$\mathrm{d}$'s as default, see §\ref{sec:ChangingDefaults}
on changing default settings. As noted earlier, upright `d's conform
to the standard ISO 80000-2 and are used in this document.
\end{itemize}
For inclusion in a line of text you might prefer to use a slash-fraction
form of derivative. That is achieved with the \verb`\difs` command:
\verb`$ \difs yx $` produces $ \difs yx $. If you want still more
compactness, you can use the \verb`\difc` (`c' for \emph{compact})
command: \verb`$ \difc yx $` produces the form $ \difc yx $.

Partial derivatives follow the same pattern as ordinary derivatives.
The commands this time are \texttt{\textbackslash}\verb`diffp`,
\verb`\difsp` and \verb`\difcp` for (upright) fraction, slash fraction
and compact forms of partial derivative. Thus \verb`\diffp{F}{x}`,
or \verb`\diffp Fx` with a saving on keystrokes, produce $\diffp Fx$
in text style and 
\[
\diffp{F}{x}
\]
in display style. (As for \verb`\diff`, the omission of braces is
possible when dealing with a single-token differentiand or differentiation
variable.) For inline use, \verb`\difsp Fx`, displays as $\difsp Fx$
and \verb`\difcp` displays as $\difcp Fx$. Given that \verb`\partial`
takes 8 keystrokes to type, all forms\emph{ }economise on keystrokes. 

\subsection{Spacing before the differentiand}

\label{subsec:Spacing-before-derivand}There are (at least) two different
ways in which we think of derivatives. We are all familiar with the
argument presented in elementary calculus books where a curve is shown,
and also a point on the curve through which a chord has been drawn.
The chord is the hypotenuse of a small right-angled triangle, the
other sides having lengths $\delta x$ and $\delta y$ and being parallel
to the coordinate axes. The slope of the chord is $\diff.gd.yx$.
By drawing smaller and smaller chords through the point, the ratio
$\diff.gd.yx$ approaches the slope of the tangent to the curve at
the point. We write 
\[
\diff yx
\]
for the limit of $\diff.gd.yx$. It is natural following this line
of argument to think of $\dl y$ and $\dl x$ as tiny lengths, like
$\delta y$ and $\delta x$, in which case it would be quite wrong
to insert space between the $\mathrm{d}$ and the $y$ (let alone
the $\mathrm{d}$ and the $x$). $\dl y$ is a single object, called
a differential, and we write expressions like 
\[
\dl y=\diff yx\dl x
\]
and justly call the fraction in this expression a \emph{differential
coefficient}. 

But there is another way of viewing differentiation: as a process
producing (or \emph{deriving}) one function, $y'(x)$, from another,
$y(x)$. Here the sense is of applying $\diff{}x$ to a quite separate
object, the function $y(x)$. Although we include $y(x)$ in the numerator
it is distinct from the $d$ and should be separated from it by a
small space:\footnote{I thank \noun{Hans Schülein} for first raising this issue with me
and for subsequent thoughtful comments.}
\[
y'(x)=\diff{y(x)}x.
\]
Here the fraction on the right is another name for the derived function
$y'$ and is justly called the \emph{derivative} of $y$. As you can
see a small space has been inserted between the $\mathrm{d}$ and
the $y$ in the numerator. By default the space is \verb`3 mu` but
with the ability to stretch by \verb`1 mu` or shrink by\texttt{ }\verb`2 mu`
--\texttt{ }\verb`3 mu plus 1 mu minus 2 mu`\texttt{ }in \TeX -speak\footnote{Or even \texttt{3muplus1muminus2mu}.}
-- as \TeX{} adjusts lines to fit on the page. (A `mu' is a `math
unit' and is one eighteenth of a quad.) The size of the space inserted
by default can be easily changed; see §\ref{sec:Variant-forms} and
§\ref{sec:ChangingDefaults}. 
\begin{itemize}
\item You may want all or most of your derivatives to have this space before
the derivand. The \verb`spaced=1` package option switches this behaviour
on. However, I have used the \verb`spaced=-1` option for the present
document which inserts space only if the derivand contains \emph{more
than one token}. Thus \verb`y(x)` will have space inserted before
it, but \verb`y` alone will not. This (I think) maintains the distinction
between a differential coefficient, thought of as a ratio of tiny
lengths, and a derivative, thought of as an operator applied to a
function. \verb`spaced=0` inserts no space before the derivand.
\item In version 4 of \verb`diffcoeff` an argument was added to the \verb`\diff`
command to manually introduce a space (the \verb`!` switch) before
the differentiand. This was a mistake and has been removed. If you
wish to adjust the spacing, there are plenty of (short) commands in
\LaTeX{} and \verb`diffcoeff` to do the job; see immediately below,
§\ref{subsec:Spacing-commands}.
\end{itemize}
Slash-form derivatives also allow space before the derivand. By default
this is \verb`2 mu plus 1 mu minus 2 mu`, slightly reduced from the
fraction-form value to avoid visually detaching the initial `d'
operator from the derivative as a whole. The value can be changed;
see §\ref{sec:Variant-forms} and §\ref{sec:ChangingDefaults}. For
the present document, with \verb`spaced=-1`, multi-token derivands
have the space inserted, single-token derivands do not:
\begin{centred}
\verb`$ \difs{\ln\sin x}x, \quad \difs st $` $\Longrightarrow$ $ \difs{\ln\sin x}x,  \quad \difs st $.
\end{centred}
For \emph{compact-form} derivatives the space before the derivand
is \emph{always} inserted, irrespective of the setting of the \verb`spaced`
package option, since the subscript precludes the entire symbol ever
being viewed as a differential -- it is always an operator operating
on a function. The inserted space, \verb`1 mu plus 1 mu minus 1 mu`
by default, can be changed should you wish; see §\ref{sec:Variant-forms}
and §\ref{sec:ChangingDefaults}:
\begin{centred}
\verb`$ \difc{\ln\sin x}x, \quad \difc st $` $\Longrightarrow$ $ \difc{\ln\sin x}x, \quad \difc st $.
\end{centred}
(The space is less for compact forms since the subscript already provides
some visual separation.) The \verb`spaced` package option has the
same effects on partial derivatives. Thus with \verb`spaced=1` or
\verb`spaced=-1`, \verb`3 mu plus 1 mu minus 2 mu` of space is inserted
before the differentiand \verb`F(x,y)` in the first member of the
following example, space of \verb`2 mu plus 1 mu minus 2 mu` in the
second, and space of \verb`1 mu plus 1 mu minus 1 mu` in the third:
\begin{centred}
\verb`\[ \diffp{F(x,y)}x,\;\difsp{F(x,y)}x,\;\difcp{F(x,y)x, \]`
\end{centred}
$\Longrightarrow$ \[ \diffp{F(x,y)}x,\;\difsp{F(x,y)}x,\;\difcp{F(x,y)}x. \]

\noindent But for single-token differentiands in this document the
space is not inserted for upright and slash-form derivatives:
\begin{centred}
\verb`\[ \diffp Fx,\quad \difsp Fx. \]` $\Longrightarrow$ \[ \diffp Fx,\quad \difsp Fx. \]
\end{centred}
If you always want the space present, use \verb`spaced=1`; if you
never want the space for upright- or slash-form derivatives, or wish
to insert such space always `by hand', use \verb`spaced=0`.

\subsubsection{Spacing commands}

\label{subsec:Spacing-commands}\LaTeX{} has its own explicit spacing
commands. In particular \verb`\,` which is \verb`3 mu` (a thin space)
and \verb`\!` which is \verb`-3 mu` (a negative thin space) are
convenient in math mode. The \verb`diffcoeff` package adds four simple
spacing commands to `fill in (most of) the gap' between these two.
These are 
\begin{description}
\item [{\texttt{\textbackslash negmu}}] insert spacing of \verb`-1 mu`; 
\item [{\texttt{\textbackslash nilmu}}] insert spacing of \verb`0 mu`
(cf. use of an empty brace pair \texttt{\textbf{\{\}}} );
\item [{\texttt{\textbackslash onemu}}] insert spacing of \verb`1 mu`; 
\item [{\texttt{\textbackslash twomu}}] insert spacing of \verb`2 mu`. 
\end{description}
It is also worth recalling here the reduced spacing around scalable
delimiters that results from using \verb`\mleft`, \verb`\mright`
in place of \verb`\left`, \verb`\right`; see §\ref{sec:Package-options}
for the \verb`mleftright` package option, and the example at §\ref{subsec:mleftright}.

\subsection{Higher order derivatives}

\label{subsec:Higher-order-derivatives}An optional argument allows
the order of differentiation to be specified. The order need not be
a number; an algebraic order of differentiation is perfectly acceptable
as is a mix of the two:
\begin{centred}
\verb`\[ \diff[2]yx, \quad \diff[n+1]yx. \]` $\Longrightarrow$ \[ \diff[2]yx, \quad \diff[n+1]yx. \]
\end{centred}
As mentioned, the braces can be and have been omitted around the $x$
and $y$ since they are single tokens. The square brackets around
the optional order-of-differentiation argument are essential. In slash
form, 
\begin{centred}
\verb`$ \difs[2]yx, \quad \difs[n+1]yx $` $\Longrightarrow$ $ \difs[2]yx, \quad \difs[n+1]yx, $
\end{centred}
the latter of which is a bit of an eyesore. In compact form, 
\begin{centred}
\verb`$ \difc[2]yx,\quad\difc[n+1]yx $` $\Longrightarrow$ $\difc[2]yx,\quad\difc[n+1]yx$.
\end{centred}
Note that entering \verb`1` as the optional argument has no effect: 
\begin{centred}
\verb`$ \diff[1]yx,\;\difs[1]yx,\;\difc[1]yx $` $\Longrightarrow$
$\diff[1]yx,\;\difs[1]yx,\;\difc[1]yx $.
\end{centred}
For partial derivatives when differentiating in only one variable
the pattern is the same:
\begin{centred}
\verb`\[ \diffp[2]yx, \quad \diffp[n+1]yx. \]` $\Longrightarrow$
\[ \diffp[2]yx, \quad \diffp[n+1]yx. \]
\end{centred}
For the slash forms, 
\begin{centred}
\verb`$ \difsp[2]yx,\quad \difsp[n+1]yx $` $\Longrightarrow$ $ \difsp[2]yx,\quad \difsp[n+1]yx. $
\end{centred}
and in compact form,
\begin{centred}
\verb`$ \difcp[2]yx,\quad \difcp[n+1]yx $` $\Longrightarrow$ $ \difcp[2]yx,\quad \difcp[n+1]yx. $
\end{centred}
For partial differentiation in more than one variable -- so-called
\emph{mixed} partial derivatives -- see §\ref{subsec:Mixed-partial-derivatives}. 

\subsection{Appending the differentiand}

\label{subsec:Appending-the-differentiand}Some differentiands are
too big or awkward to be placed neatly in the numerator of a derivative
and it is natural to \emph{append} them to a preceding differential
operator. One could leave the numerator argument empty in the \verb`\diff`
or \verb`\diffp` command and follow the command with the differentiand,
but \verb`diffcoeff` offers a better way: star the \verb`\diff`
or \verb`\diffp` command. This tells \verb`diffcoeff` to append
the differentiand. Thus suppose the differentiand is a polynomial,
say $ax^{2}+bx+c$. Add a star (an asterisk) to the \verb`\diff`
command:
\begin{centred}
\verb`\[ \diff*{(ax^2+bx+c)}x \]` $\Longrightarrow$ \[\diff*{(ax^{2}+bx+c)}x.\]
\end{centred}
Or, for a partial derivative, one might want to indicate in the differentiand
all the variables on which it depends:
\begin{centred}
\verb`\[ \diffp*[2]{\Phi(x,y,z)}x \]` $\Longrightarrow$ \[ \diffp*[2]{\Phi(x,y,z)}x. \]
\end{centred}
A virtue of using an asterisk to append the differentiand is that
if one isn't sure whether a differentiand should be appended or not,
it is an easy matter to simply insert or delete the asterisk to compare
the results. 

For instance, a second derivative is an iterated derivative -- one
in which a derivative forms the differentiand of another . Thus
\begin{centred}
\verb`\[ \diff[2]yx = \diff*{\diff yx}x \]` $\Longrightarrow$ \[ \diff[2]yx=\diff*{\diff yx}x .\]
\end{centred}
\noindent This result is more elegant to my eye than what results
when removing the asterisk,
\begin{centred}
\noindent \verb`\[ \diff[2]yx = \diff{\diff yx}x \]` $\Longrightarrow$
\[ \diff[2]yx=\diff{\diff yx}x ,\]
\end{centred}
\noindent although whether the \emph{meaning} is clearer is moot.

Since the differentiand is appended \emph{by default} in compact-form
derivatives, starring such a derivative has no effect other than when
a second asterisk is used to transpose the order of arguments.

\subsubsection{Transposing the argument order}

\label{subsec:Transposing-the-argument}If a \emph{second} asterisk
follows the first, the order of the arguments specifying the differentiand
on the one hand and variable or variables of differentiation on the
other are reversed. Thus it is clearer to the eye to write
\begin{centred}
\verb`\[ \diff**x{(ax^2+bxy+cy^2)} \]` $\Longrightarrow$ \[\diff**x{(ax^2+bxy+cy^2)}\] 
\end{centred}
than \verb`\[ \diff*{(ax^2+bxy+cy^2)}x \]`, where the eye has to
search for the variable of differentiation. This is especially the
case if the differentiand contains more than one variable and commands
like \verb`\frac` or \verb`\sqrt` requiring braced arguments:
\begin{centred}
\verb`\[ \diffp**x{\frac1{\sqrt{x^2-y^2}}} \]` $\Longrightarrow$
\[ \diffp**x{\frac1{\sqrt{x^2-y^2}}} \]
\end{centred}
For compact-form derivatives the initial, appending asterisk is always
implicitly present. However, it must be \emph{explicitly} present
for the second asterisk to take effect:
\begin{centred}
\verb`\[ \difcp yx,\quad \difcp*yx,\quad \difcp**yx \]` $\Longrightarrow$
\[ \difcp yx,\quad \difcp*yx,\quad \difcp**yx \]
\end{centred}

\subsubsection{Operator parenthesizing}

In slash style with the star option, the polynomial example becomes
\begin{centred}
\verb`$ \difs*{(ax^2+bx+c)}x $` $\Longrightarrow$ $ \difs*{(ax^2+bx+c)}x $
\end{centred}
\noindent where parentheses have been automatically inserted around
the differential operator. Similarly, for slash-style partial derivatives,
\begin{centred}
\verb`\( \difsp*[n]{f(x)}x \)` $\Longrightarrow$ \( \difsp*[n]{f(x)}x \)
\end{centred}
\noindent parentheses are again inserted automatically around the
differential operator. Like other elements of automatic formatting,
this behaviour is user-adjustable; see §§\ref{sec:Variant-forms},
\ref{sec:ChangingDefaults}.

\subsection{Point of evaluation/variables held constant}

\label{subsec:Point-of-evaluation}If you want to specify a point
at which a derivative is evaluated, append a final optional argument:
\begin{centred}
\verb`\[ \diff[2]yx[0] \]` $\Longrightarrow$ \[ \diff[2]yx[0] \]
\end{centred}
\noindent Note that there must be \emph{no space} before the left
square bracket of the trailing argument, otherwise it will be treated
as part of the wider mathematical expression of which the derivative
is part and typeset as such. (This should not cause a \LaTeX{} error.) 
\begin{itemize}
\item If you prefer to use subscripted \emph{square} brackets 
\[
\diffp.].{F(x,y)}x[(0,0)]
\]
or a subscripted vertical rule after the derivative
\[
\diffp.|.{F(x,y}x[(0,0)]
\]
to indicate a point of evaluation, then this can easily be done; see
specifically §\ref{subsec:variantPoint-of-evaluation} (or §\ref{sec:ChangingDefaults}
on changing default settings). Parentheses are the ISO recommendation;
see ISO 80000-2. 
\end{itemize}
Because the slash form spreads the derivative out horizontally, parentheses
are the natural way in this case to indicate a point of evaluation:
\begin{centred}
\verb`$ \difs{\ln sin x}{sin x}[x=\pi/3] $` $\Longrightarrow$ $ \difs{\ln\sin x}{\sin x}[x=\pi/3]$.
\end{centred}
\noindent A vertical rule can easily become too remote from the opening
$\mathrm{d}$ of the differential coefficient: $\difs.|.{\ln\sin x}{\sin x}[x=\pi/3]$.
Parentheses tie the whole cluster of symbols together.

One reason to query the ISO preference for subscripted parentheses
to indicate a point of evaluation is that subscripted parentheses
are used with partial derivatives to indicate variables held constant.
This occurs frequently in thermodynamics for example. In the following
well-known relation in thermodynamics, the differentiands are appended
and the trailing argument is used to indicate the variables held constant: 
\begin{centred}
\verb`\[ \diffp*{\frac PT}U[V] = \diffp*{\frac 1T}V[U] \]` $\Longrightarrow$
\[ \diffp*{\frac PT}U[V] = \diffp*{\frac 1T}V[U]. \]
\end{centred}
\noindent This is much easier to write than building the expressions
`by hand', starting with \verb`\left(` and finishing with \verb`_U`.

\subsubsection{Superscripts}

It is easy to add a superscript to a derivative to indicate evaluation
at two points and the difference between the values:
\begin{centred}
\verb`\[ \diff {\sin x}x[0]^{\!\!\pi/2} \]` ${\displaystyle \Longrightarrow}$
\[ \diff{\sin x}x[0]^{\!\!\pi/2} \]
\end{centred}
but to my eye either square brackets or a vertical rule are clearer
for this purpose (and do not involve nudging the subscript or superscript
closer to the right delimiter); see §\ref{sec:Variant-forms}.

\subsubsection{Empty trailing argument}

\noindent If the trailing argument is included but left empty it will,
with the default set-up, wrap the derivative in parentheses but with
no subscript. This fact can be exploited. Thus, for a particle of
mass $m$ moving along a line, distance $x$ at time $t$, the kinetic
energy is:
\begin{centred}
\noindent \verb`$ \tfrac 12 m \difs xt[]^2 $` $\Longrightarrow$
$\tfrac{1}{2}m\difs xt[]^{2}$.
\end{centred}
Or, again exploiting the parentheses resulting from an empty trailing
argument, Lagrange's equations of motion in analytic mechanics can
be written,
\begin{centred}
\verb`\[ \diffp L{q_k}-\diff**t{ \diffp L{\dot{q}_k}[] } = 0 \]`
$\Longrightarrow$\[\diffp L{q_k}-\diff**t{\diffp L{\dot{q}_k}[]} = 0.  \]
\end{centred}
\noindent (See §\ref{subsec:Transposing-the-argument} for the double
asterisk.) Like the author, you may feel that there is too much whitespace
between $\diff{}t$ and the left parenthesis in this expression. One
obvious remedy is to insert a negative thin space \verb`\!` before
the second \verb`\diffp` command. Another is to use the package \verb`mleftright`.

\subsubsection{Use of the package \texttt{mleftright}}

\label{subsec:mleftright}The \verb`mleftright` package `tightens'
the spacing around \verb`\left`, \verb`\right` delimiters. The user
either by explicitly replaces \verb`\left`, \verb`\right` by \verb`\mleft`
and \verb`\mright` or uses the command \verb`\mleftright` which
effectively turns subsequent occurrences of \verb`\left`, \verb`\right`
into \verb`\mleft`, \verb`\mright`; \verb`\mleftrightrestore` returns
\verb`\left`, \verb`\right` to their original selves. This process
can be `short-circuited' by using the \verb`diffcoeff` package
option \verb`mleftright` that inserts \verb`\mleftright` in the
preamble; see §\ref{sec:Package-options}.

However, that package option is not used in this document. Nonetheless
\verb`mleftright` is a required package of version 5 of \verb`diffcoeff`
and its commands are available for use. Hence to reduce the whitespace
I can write
\begin{verbatim}
    \mleftright
    \[ \diffp L{q_k}-\diff**t{\diffp L{\dot{q}_k}[]} = 0. \]
    \mleftrightrestore
\end{verbatim}
$\Longrightarrow$ \mleftright
\[ \diffp L{q_k}-\diff**t{\diffp L{\dot{q}_k}[]} = 0. \]
\mleftrightrestore

\noindent which is better, but better still to my eye is to also add
a negative thin space \verb`\!` before the second \verb`\diffp`
command:
\begin{verbatim}
    \mleftright
    \[ \diffp L{q_k}-\diff**t{ \!\diffp L{\dot{q}_k}[] } = 0 \]
\end{verbatim}
\noindent $\Longrightarrow$ \mleftright\[\diffp L{q_k}-\diff**t{\!\diffp L{\dot{q}_k}[]} = 0.  \]

\noindent The problem is that in addition to the space around \verb`\left`,
\verb`\right` pairs (which is reduced by issuing the command \verb`\mleftright`),
there is also the space inserted by \verb`diffcoeff` before a multi-token
differentiand. The user should be aware of this, and may wish to define
a `variant form' (see §\ref{sec:Variant-forms}) that introduces
no space before an appended differentiand.

\subsection{Mixed partial derivatives}

\label{subsec:Mixed-partial-derivatives}The new thing with partial
derivatives, not present with ordinary derivatives, is so-called \emph{mixed}
partial derivatives, where differentiation occurs in more than one
variable. If each variable is differentiated only to the first order,
then it is easy to specify the derivative. Suppose $F$ is a function
of three variables, $x$, $y$ and $z$. Then
\begin{centred}
\verb`\[ \diffp F{x,y,z},\quad\diffp{F(x,y,z)}{x,y,z}. \]` $\Longrightarrow$
\[ \diffp F{x,y,z},\quad \diffp{F(x,y,z)}{x,y,z}. \]
\end{centred}
(The \verb`spaced=-1` package option inserts space before the multi-token
differentiand in the second of these.)

The variables of differentiation are listed in order in a comma list
-- \verb`{x,y,z}` -- forming the second mandatory argument. The
total order of differentiation (3 in this example) was inserted automatically.
It did not need to be specified or calculated `by hand' -- \verb`diffcoeff`
did the calculation. 

The slash form is
\begin{centred}
\verb`$ \difsp F{x,y,z} $` $\Longrightarrow$ $ \difsp F{x,y,z}$,
\end{centred}
as expected, and the compact form is 
\begin{centred}
\verb`$ \difcp F{x,y,z} $` $\Longrightarrow$ $ \difcp F{x,y,z}$.
\end{centred}
One might wonder about even more compact notations like $\partial_{xyz}^{\onemu3}F$
for this example but it becomes messy if different orders of differentiation
are involved for different variables.

To differentiate variables to higher order, their orders need to be
specified explicitly. To do so use a comma list for the optional argument:
\begin{centred}
\verb`$ \difcp[2,3]F{x,y,z} $` $\Longrightarrow$ $ \difcp[2,3] F{x,y,z}$,

\verb`\[ \diffp[2,3]F{x,y,z} \]` $\Longrightarrow$ \[ \diffp[2,3]F{x,y,z}. \]
\end{centred}
\noindent Notice that the overall order of the derivative -- 6 --
in the second of these is again automatically calculated and inserted
as a superscript on the $\partial$ symbol in the numerator. 

In the example, the comma list of orders has only \emph{two} members,
although there are \emph{three} variables. It is assumed that the
orders given in the comma list apply in sequence to the variables,
the first order to the first variable, the second to the second variable,
and so on, and that any subsequent orders not listed in the optional
argument are, by default, 1. Thus we need to specify only 2 and 3
in the example; the order of differentiation of $z$ is 1 by default.
But you \emph{cannot} use an order specification like \verb`[,,2]`;
instead write {\ttfamily\verb`[1,1,2]`} (which is the natural
thing to do in any case).\textbf{ }It is only the \emph{tail} of an
order specification which can be omitted.

In the other direction, if there are more orders of differentiation
specified than there are variables, the list of orders is truncated
to match the number of variables. 

\subsubsection{Algebraic orders of differentiation}

Orders of differentiation do not need to be numerical. They can also
be algebraic:
\begin{centred}
\verb`\[ \diffp[2m-1,m+1,2]F{x,y,z} \]` $\Longrightarrow$ \[ \diffp[2m-1,m+1,2]F{x,y,z}\]
\end{centred}
The total order of differentiation is still calculated by \verb`diffcoeff`.
Or again,
\begin{centred}
\verb`\[ \diffp[1,km+1,m+k-1]{F(x,y,z)}{x,y,z} \]` $\Longrightarrow$
\[ \diffp[1,km+1,m+k-1]{F(x,y,z)}{x,y,z}. \]
\end{centred}

\subsubsection{Order-override command: \texttt{\textbackslash difoverride}}

\noindent \label{subsec:Order-override-command}$m+k+km+1$ factorizes
to $(k+1)(m+1)$ and you may prefer to express the total order of
differentiation in this form. \verb`diffcoeff` is not a computer
algebra system and does not do such factorizations but the \verb`\difoverride`
command offers the means to use the factorized form as the total order
of differentiation in the derivative. (The \verb`\difoverride` command
replaces the order-override option of the \verb`\diff` command in
version 4 of \verb`diffcoeff`. That option is no longer available.)
\begin{verbatim}
    \[
      \difoverride{(k+1)(m+1)}
      \diffp[1,km+1,m+k-1]{F(x,y,z)}{x,y,z}
    \]
\end{verbatim}
$\Longrightarrow$     \[
      \difoverride{(k+1)(m+1)}
      \diffp[1,km+1,m+k-1]{F(x,y,z)}{x,y,z}
    \]

\verb`\difoverride` takes only one (mandatory) argument, the total
order of differentiation, which it stores. When \verb`\difoverride`
is non-empty, then the algorithm that calculates the total order is
sidestepped. It does not get called at all. In this way one can present
not only the total order in whatever manner one wishes but essentially
arbitrary material as a superscript to the $\partial$ symbol in the
numerator. (For compact-form derivatives, which do not use a total
order of differentiation, the command is irrelevant.)

Note that in the example \verb`\difoverride` has been used \emph{within
}the math environment. This is good practice. It prevents the contents
of \verb`\difoverride` erroneously overriding the orders of later
derivatives: 
\begin{verbatim}
    $ \difoverride{N} \difsp[m,n]f{x,y} $,\quad
    $ \difsp[1,2]f{x,y} $.
\end{verbatim}
$\Longrightarrow$     $ \difoverride{N} \difsp[m,n]f{x,y} $,\quad
    $ \difsp[1,2]f{x,y} $.

\noindent There is no spillover here whereas by placing the override
command outside the math environment the command reaches beyond the
intended target. To cancel the overreach, I have used \verb`\difoverride`
with an \emph{empty} argument, :
\begin{verbatim}
    \difoverride{K} 
    $ \difsp[m,n]f{x,y} $,\quad
    $ \difsp[1,2]f{x,y} $,\quad
    \difoverride{} 
    $ \difsp[1,2]f{x,y}$.
\end{verbatim}
$\Longrightarrow$     \difoverride{K} $ \difsp[m,n]f{x,y} $,\quad
    $ \difsp[1,2]f{x,y} $,\quad
    \difoverride{} $ \difsp[1,2]f{x,y}$.

\noindent Clearly one should use the override command \emph{within}
the math environment.

\subsubsection{Parentheses}

Auto-calculation of the total order accommodates the simple use of
parentheses:
\begin{centred}
\verb`\[ \diffp[2m-(k+1),2(k+1)-m]{F(x,y,z)}{x,y,z} \]` $\Longrightarrow$
\[ \diffp[2m-(k+1),2(k+1)-m]{F(x,y,z)}{x,y,z} \]
\end{centred}
This is an example of the use of \emph{dynamic} parentheses: the left
parenthesis in each case is preceded by a number or a sign. In evaluating
the total order \verb`diffcoeff` multiplies out the expression (or
that is the effect).

On the other hand, an order specification like \verb`[f(n+1),f(n-1)]`
is an example of the use of \emph{static} parentheses where they are
part of the familiar `function of' notation -- in this case a function
$f$ of some variable, say $x$, evaluated at $x=n\pm1$. \verb`diffcoeff`
\emph{always} interprets a left parenthesis preceded by something
that is neither number nor sign in this way. It does not try to multiply
out such expressions when calculating the total order.

The following example combines both uses -- and includes a nested
pair of (dynamic) parentheses:
\begin{centred}
\verb`\[\diffp[2(f(n)-(m-1)),5-(f(n)+m)]F{x,y}\]` $\Longrightarrow$
\[\diffp[2(f(n)-(m-1)),5-(f(n)+m)]F{x,y}\]
\end{centred}
Where confusion arises is with specifications like \verb`[m(k-1)+1,m(k+1)-1]`
where \verb`m` could be interpreted as either a function or a variable.
As stated, \verb`diffcoeff` \emph{always} interprets a left parenthesis
preceded by something that is not a number or a sign as signalling
`function of'. Hence:
\begin{centred}
\verb`\[\diffp[m(k-1)+1,m(k+1)-1]F{x,y}\]` $\Longrightarrow$ \[\diffp[m(k-1)+1,m(k+1)-1]F{x,y}\]
\end{centred}
\noindent If, in fact, \verb`m` is intended as a \emph{variable}
then the order-override command is there to rescue the situation:
\begin{verbatim}
    \[
      \difoverride{2mk}
      \diffp[m(k-1)+1,m(k+1)-1]F{x,y}
    \]
\end{verbatim}
$\Longrightarrow$     \[
      \difoverride { 2mk }
      \diffp[m(k-1)+1,m(k+1)-1]F{x,y}
    \]

\subsubsection{Error messages}

The order-override command is also needed when calculation of the
total order is beyond the abilities of \verb`diffcoeff`. The package
is \emph{not} a computer algebra system. It can cope with order specifications
where variables are followed by diverse arithmetic operators: \verb`n^2`,
\verb`m\times n`, \verb`m/2` and the like cause no problems. But
a \emph{number} can be followed \emph{only} by a sign or a variable
or a left parenthesis. Anything beyond this will raise an error. For
instance
\begin{centred}
\verb`\[ \diffp[2^k]F{x,y} \]` 
\end{centred}
produces a message beginning `\texttt{! Package diffcoeff Error:}'
and continuing,
\begin{verbatim}
    number followed by ^ in the order spec. [2^k,1] on
    line xx. Calculation of the total order of 
    differentiation fails in this case. Use the 
    \difoverride command to enter the total order. 
    See the diffcoeff documentation for further
    information.
\end{verbatim}
(The \verb`xx` will be replaced by a specific line number in each
case. Line breaking may also differ from case to case.) To avoid such
errors and enable compilation to proceed, use the override command:
\begin{verbatim}
    \[ 
      \difoverride { 2^{n+1} }
      \diffp[2^n+1,2^n-1]F{x,y} 
    \]
\end{verbatim}
$\Longrightarrow$     \[ 
      \difoverride { 2^{n+1} }
      \diffp[2^n+1,2^n-1]F{x,y} 
    \]

There are limitations on what order specifications the \verb`diffcoeff`
package can `digest', but in real life that is unlikely to be significant.
Mixed partial derivatives are used far less often than the pure derivatives,
and when they \emph{are} used it is nearly always to low numerical
orders like 1 or 2. For those rare other cases, \verb`\difoverride`
is always available. 

\subsubsection{Comma list of variables of differentiation}

\label{subsec:Comma-list-of-vars}In tensor calculus differentiations
are almost always in terms of super- or subscripted coordinates. In
many other contexts this is the case too -- the reciprocal of the
temperature in thermodynamics or generalized coordinates in analytical
mechanics. This is why a comma list is used in \verb`diffcoeff` for
specifying variables of differentiation for mixed partial derivatives.
Although it would be nice to write the minimal \verb`{xy}` rather
than \verb`{x,y}` when two variables $x$ and $y$ are involved,
the extra writing is trivial and the comma list allows a simpler handling
of multi-character variables. For instance in tensor calculus we get
expressions like
\begin{centred}
\verb`\[ \diffp{A_i}{x^j,x^k} \]` $\Longrightarrow$ \[ \diffp{A_i}{x^j,x^k}. \]
\end{centred}
\noindent It is easier to write \verb`{x^j,x^k}` here than, say,
\verb`{{x^j}{x^k}}` to distinguish the variables. It's also easier
to read, particularly if the indices themselves get ornamented and
need surrounding braces:
\begin{centred}
\verb`\[ \diffp{A_i}{x^{j'},x^{k'}} \]` $\Longrightarrow$ \[ \diffp{A_i}{ x^{j'},x^{k'} }. \]
\end{centred}
\noindent Compare that variable specification with \verb`{{x^{j'}}{x^{k'}}}`.
Admittedly some extra whitespace would help here, but the point stands:
the comma list requires fewer nested braces -- unless a variable
of differentiation includes a comma, for then the comma needs to be
enclosed in braces. There are plenty of instances of this out in the
world (see, e.g., the last equation of (\ref{eq:eg1})) but it is
overall a rare occurrence.

\subsubsection{Spacing in the denominator}

\label{subsec:Minutiae-of-spacing}In Chapter 18 of the \emph{The
\TeX book}, Knuth suggests inserting a thin space, {\ttfamily\verb`\,`}
(or {\ttfamily\verb`3 mu`}), between differentials in appropriate
contexts, giving as an example $dx\,dy=r\,dr\,d\theta$. In the denominator
of a derivative, however, that degree of extra spacing -- to my eye
-- seems too great, interfering with seeing the derivative `as a
whole', 
\[
\diffp.3mu.F{x,y,z},
\]
especially for the slash-form of derivative: $\difsp.3mu.F{x,y,z}$.
Some spacing is desirable, but less. By default \verb`diffcoeff`
inserts \verb`2 mu` (with stretch and shrink) between the differentials:
$\difsp F{x,y,z}$.

Should a differentiation occur to higher order and so a variable acquire
a superscript, an adjustment is made to the extra spacing. By default
\verb`1 mu` is subtracted from the default spacing. Thus in 
\[
\diffp[2]F{x,y,z},
\]
\verb`2 mu` of spacing is inserted between the $\partial y$ and
$\partial z$, but because the superscript already provides some separation
between them, only \verb`1 mu` is inserted between $\partial x^{2}$
and $\partial y$. The values used for the spacing and its adjustment
in the presence of a superscript can be changed by the user; see Chapter~\ref{chap:Templates-defaults-variants}. 

When the variables themselves are super- or subscripted, as happens
in tensor calculus, no automatic adjustment is made. Any fine-tuning
must be done by the user using explicit spacing commands -- like
\verb`\negmu` (a space of \verb`-1 mu`); see §\ref{subsec:Spacing-commands}:
\begin{centred}
\verb`\[ \diffp{A_i}{ x^j\negmu,x^k } \]` $\Longrightarrow$ \[ \diffp{A_i}{ x^j\negmu,x^k }.\]
\end{centred}
\noindent The \verb`\negmu` decreases the spacing between the terms
from the default \verb`2 mu` (with stretch and shrink) to \verb`1 mu`.

\subsection{Multi-token variables: parenthesizing}

Differentiating a function of a function may involve a multi-character
differentiation variable. For instance, to differentiate $\ln\sin x$
in $x$ means forming the product
\begin{centred}
\verb`\[ \diff{\ln\sin x}{\sin x}\diff{\sin x}x \]` $\Longrightarrow$
\[\diff{\ln\sin x}{\sin x}\diff{\sin x}x.\]
\end{centred}
Forming the \emph{second} derivative of $\ln\sin x$ will now involve
forming, among other quantities,
\begin{centred}
\noindent \verb`\[\diff[2]{\ln\sin x}{\sin x}\]` $\Longrightarrow$
\[\diff[2]{\ln\sin x}{\sin x}\]
\end{centred}
\noindent Parentheses have been inserted automatically by \verb`diffcoeff`
around $\sin x$ in the denominator to avoid any visual hint that
we are differentiating in the sine of $x^{2}$.

\emph{That} is the problem: with a long (multi-character) variable,
the superscript in a higher order derivative may look as if it applies
to only part -- the last character -- of a multi-character variable.
To solve that problem, \verb`diffcoeff` inserts parentheses around
the variable -- for \emph{higher-order} derivatives, but not for
first-order derivatives where the problem does not arise. You may
prefer \verb`diffcoeff` not to parenthesize by default. Changing
the default setting is easily accomplished; see §\ref{sec:ChangingDefaults}.

And if you do want parentheses, are they in the right place? Logically,
no. They should include the $d$: $(d\sin x)^{2}$ -- it is the differential
$d\sin x$ that is of the second order. But as the examples in the
Rogues' Gallery show -- see particularly (\ref{eq:eg3}) -- the
inclination seems to be to do otherwise. This may be because one wants
in any case to parenthesise the variable to show that the `d' symbol
attaches to the whole variable and not just its \emph{first} character.
A second, outer pair of parentheses then seems too fussy and detracts
from comprehending the symbol `at a glance': 
\[
\diff.(dv).[2]{f(x)}{(x/k)}.
\]
Customary but illogical notations are familiar in mathematics --
think of the position of the superscripts in an identity like $\sin^{2}\theta+\cos^{2}\theta=1$.
In any case,  the manner of this wrapping in parentheses -- if any
-- of long variables for \emph{higher order} derivatives is customisable
(§\ref{sec:ChangingDefaults}).

For first order derivatives parenthesising does not occur automatically.
If you want the variable of differentiation to be parenthesised, you
need to do it yourself:
\begin{centred}
\verb`\[\diff {f(x)}{x/k}, \quad \diff {f(x)}{(x/k)}.\]` $\Longrightarrow$
\[ \diff {f(x)}{x/k}, \quad\diff {f(x)}{(x/k)}.\]
\end{centred}
To my eye, in this particular case, the parenthesized version seems
necessary. The discussion applies equally to ordinary and partial
derivatives. In thermodynamics and statistical mechanics one may want
to differentiate in the reciprocal of the temperature, $1/\Theta$
say:
\begin{centred}
\verb`\[ \diffp[2]q{\frac 1\Theta} \]` $\Longrightarrow$ \[ \diffp[2]q{\frac 1\Theta}.\]
\end{centred}
\noindent As noted, when  differentiating to first order, parenthesising
is up to the user:
\begin{centred}
\verb`\[ \diffp q{(\frac 1\Theta),V} \]` $\Longrightarrow$ \[ \diffp q{(\frac 1\Theta),V}. \]
\end{centred}

\chapter{Templates, defaults \& variants}

\label{chap:Templates-defaults-variants} \verb`diffcoeff` is built
on the facilities offered by the \verb`xtemplate` package (included
in the \LaTeX 3 bundle {\ttfamily\verb`l3packages`}). The
stuff of \verb`xtemplate` is \emph{templates}, their definition,
their manipulation. For \verb`diffcoeff` a template is a list of
parameter values determining how a derivative looks in the pdf. The
parameters may be broad-brush settings like whether the derivative
is built from \verb`\frac` or the slash \verb`/` or in compact form,
or whether the operator symbol is \verb`d` or \verb`\partial` (or
\verb`\nabla` or \verb`\delta` or \ldots ), or the parameters may
be finer-grained, determining minutiae of spacing, easily missed at
a casual glance but giving some cumulative overall effect.\emph{ }Access
to the parameters is gained through the command \verb`\difdef`\footnote{In version 5; it has \emph{three} arguments and replaces the two-argument
command \texttt{\textbackslash diffdef} in version 4 of \texttt{diffcoeff}.}, one argument of which is a \emph{key=value} list\emph{ }of parameter
values. Each such list is given a name (the second argument of \verb`\difdef`)
and is `turned into a derivative' by placing the name between dots
as the first argument of the appropriate \verb`\difx`, \verb`\difxp`
commands\footnote{And -- see Chapter~\ref{chap:Differentials-and-jacobians} -- of
the differential and jacobian commands, \texttt{\textbackslash dl}
and \texttt{\textbackslash jacob}.}. All this is discussed in §\ref{sec:Variant-forms} below.

\section{Template structure}

To write a derivative one doesn't want to have to type a long list
of \emph{key=value} statements each time. The \emph{default} values
given to keys is crucial. Only some of the defaults appropriate for,
say, an upright fraction ordinary derivative are going to be relevant
for a slash-fraction partial derivative let alone a compact form partial
derivative. This suggests creating a primary template as a `super-repository'
of default values and from this creating secondary or child templates
in which (only) \emph{some }of the defaults are changed -- and, if
necessary, creating from these child templates children of their own
(grandchild templates) in which again some further defaults are adjusted. 

In \verb`diffcoeff`, the template that is the `primogenitor' of
the lines of default inheritance is named \verb`DIF`. It is the repository
of all possible keys used in all possible forms of derivative (at
least in \verb`diffcoeff`!) and so has keys appropriate to upright-fraction,
slash-fraction and compact forms of derivative; it has keys appropriate
to multi-variable partial derivatives and single-variable ordinary
derivatives, but it is not actually used to form derivatives. That
is the role of its child templates \verb`DIFF`, \verb`DIFS` and
\verb`DIFC` corresponding to derivatives of upright-fraction, slash-fraction
and compact forms. These child templates inherit the defaults of \verb`DIF`
save for some settings explicitly changed in the child template relevant
to the specific forms of fraction specified by each. 

Apart from the operator symbol, most of the settings in the child
templates \verb`DIFF`, \verb`DIFS` and \verb`DIFC` are also appropriate
for partial derivatives. From a code design point of view, there is
a certain neatness at not  multiplying the number of templates in
play, but actual use -- for instance, forming the `Rogues' gallery'
of §\ref{sec:Rogues'-gallery} -- suggests the further step of creating
additional templates specifically for \emph{partial} derivatives\emph{
}in the three fraction forms. Apart from the operator symbol, the
templates \verb`DIFFP`, \verb`DIFSP` and \verb`DIFCP` inherit nearly
all the defaults of their parents \verb`DIFF`, \verb`DIFS` and \verb`DIFC`
respectively. 

Again, actual use suggests two further templates, both direct children
of \verb`DIF`, for the creation of jacobians, template \verb`DIFJ`,
and differentials, template \verb`DIFL`, with default values appropriate
to each. In all \verb`diffcoeff` uses nine templates, the arrows
in Table~\ref{tab:DIF-defaults} indicating lines of inheritance
of default values. Only the child and grandchild templates of \verb`DIF`
are used for actual construction of derivatives (and jacobians and
differentials). \verb`DIF` itself sits `above the fray'.

\begin{table}
\centering{}\caption{\protect\label{tab:Template-inheritance}Template inheritance}
\begin{tabular}{ccccc}
\multicolumn{5}{c}{\texttt{DIF}}\tabularnewline
$\hphantom{\mathtt{DIF}}$ $\swarrow$ & $\,\,\,\sswarrow$ & $\downarrow$ & $\ssearrow\,\,\,$ & $\searrow$ $\hphantom{\mathtt{IFL}}$\tabularnewline
\texttt{DIFC} & \texttt{DIFS} & \texttt{DIFF} & \texttt{DIFJ} & \texttt{DIFL}\tabularnewline
$\downarrow$ & $\downarrow$ & $\downarrow$ &  & \tabularnewline
\texttt{DIFCP} & \texttt{DIFSP} & \texttt{DIFFP} &  & \tabularnewline
\end{tabular}
\end{table}


\section{Default values for template \texttt{DIF}}

\label{sec:Default-values-DIF}Table~\ref{tab:DIF-defaults} lists
the keys available for forming derivatives and the default values
assigned in the `grandparent' template, the primogenitor, \verb`DIF`.
Different forms of derivative demand different defaults for some keys.
Where a key is relevant for more than one style of derivative the
default value is chosen according to the following precedence scheme: 
\begin{enumerate}
\item \emph{ordinary upright}-fraction derivatives in \emph{display}-style
environments
\item \emph{ordinary slash}-fraction derivatives in \emph{text}-style environments
\item \emph{ordinary compact}-form derivatives in \emph{text}-style environments
\end{enumerate}
\noindent %
\noindent\begin{minipage}[t]{1\columnwidth}%
\begin{shaded}%
\noindent Users of version 4 of \texttt{diffcoeff} will notice similarities
with and differences from the earlier version. Some key names remain
(\verb`op-symbol`), some names have changed (\verb`multi-term-sep`
for \verb`denom-term-sep`), keys beginning with an asterisk, \verb`*`,
lack a following hyphen (\verb`*derivand-sep` rather than \verb`*-derivand-sep`),
some keys have vanished (the \verb`/` keys), and there are some new
keys (\verb`lvwrap-Ldelim`, \verb`lvwrap-Rdelim`). The redesign
of the user interface -- \verb`\difs`, \verb`\difsp` for the \verb`/`
switch, the new compact form commands \verb`\difc`, \verb`\difcp`,
and bringing the jacobian and differential within the \verb`DIF`
template structure -- meant revisiting and rethinking the list of
keys. In the end it seemed simpler (less confusing) to treat this
as a completely new list rather than an amendment of the earlier one.\end{shaded}%
\end{minipage}

\medskip{}
The first column in table~\ref{tab:DIF-defaults} lists key names,
the second column default values, and the third column to which form
or forms of derivative the key is \emph{relevant --} meaning that
assigning a different value to the key can change the appearance of
the corresponding derivative in some way. The identifiers have these
meanings:
\begin{description}
\item [{f,~fp}] upright fraction ordinary derivative, partial derivative;
\item [{s,~sp}] slash fraction ordinary derivative, partial derivative;
\item [{c,~cp}] compact ordinary derivative, partial derivative;
\item [{j}] jacobian;
\item [{l}] differential.
\end{description}
In Table~\ref{tab:DIF-defaults} and following tables, all values
specifying a space require the unit (\verb`mu`) to be included; a
number alone does not suffice. (A `mu' is a `math unit', 1/18
of a quad. A thin space \texttt{\textbackslash ,} is 3 mu.) `Elastic'
spaces with stretch and shrink can be compacted, like \verb`3muplus1muminus2mu`
for \verb`3 mu plus 1 mu minus 2 mu`.

Available keys and their defaults are the following (if you are dissatisfied
with some choices, they can be changed; see the discussion at §\ref{sec:ChangingDefaults}):

\begin{table}
\centering{}\caption{{\small\texttt{DIF}} {\small defaults}\protect\label{tab:DIF-defaults}}
\begin{center}
\begin{tabular}{lrl}
\toprule 
{\small key} & {\small default} & {\small relevance}\tabularnewline
\midrule
{\small style} & {\small\texttt{frac}} & {\small f, fp, s, sp, c, cp, j}\tabularnewline
slash-tok & {\small\texttt{/}} & {\small\phantom{{\small f, fp,}} s, sp, \phantom{{\small c, cp,}}
j}\tabularnewline
slash-sep & {\small\texttt{0 mu}} & {\small\phantom{{\small f, fp,}} s, sp, \phantom{{\small c, cp,}}
j}\tabularnewline
{\small derivand-sep} & {\small\texttt{3 mu plus 1 mu minus 2 mu}} & {\small f, fp, s, sp, c, cp}\tabularnewline
{\small op-symbol} & {\small\texttt{\textbackslash mathrm\{d\}}} & {\small f, fp, s, sp, c, cp, j, l}\tabularnewline
{\small op-symbol-alt} & {\small\texttt{op-symbol}} & {\small f, fp, s, sp, \phantom{{\small c, cp,}} j}\tabularnewline
{\small op-order-nudge} & {\small\texttt{0 mu}} & {\small f, fp, s, sp, c, cp}\tabularnewline
{\small var-sup-nudge} & {\small\texttt{1 mu}} & {\small f, fp, s, sp, \phantom{{\small c, cp,} j,} l}\tabularnewline
{\small multi-term-sep} & {\small\texttt{2 mu plus 1 mu minus 1 mu}} & {\small f, fp, s, sp, c, cp, \phantom{j,} l}\tabularnewline
{\small term-sep-adjust} & {\small\texttt{-1 mu}} & {\small f, fp, s, sp, c, cp, \phantom{j,} l}\tabularnewline
{\small long-var-wrap} & {\small\texttt{d(v)}} & {\small f, fp, s, sp, \phantom{{\small c, cp,} j,} l}\tabularnewline
{\small lvwrap-Ldelim} & {\small\texttt{\textbackslash mleft (}} & {\small f, fp, s, sp, \phantom{{\small c, cp,}} j, l}\tabularnewline
{\small lvwrap-Rdelim} & {\small\texttt{\textbackslash mright )}} & {\small f, fp, s, sp, \phantom{{\small c, cp,}} j, l}\tabularnewline
{\small lvwrap-sup-nudge} & {\small\texttt{-2 mu}} & {\small f, fp, s, sp, \phantom{{\small c, cp, j,}} l}\tabularnewline
{\small outer-Ldelim} & {\small\texttt{\textbackslash left (}} & {\small f, fp, s, sp, c, cp, j, l}\tabularnewline
{\small outer-Rdelim} & {\small\texttt{\textbackslash right )}} & {\small f, fp, s, sp, c, cp, j, l}\tabularnewline
{\small elbowroom} & {\small\texttt{0 mu}} & {\small f, fp, s, sp, c, cp, j, l}\tabularnewline
{\small sub-nudge} & {\small\texttt{-5 mu}} & {\small f, fp, s, sp, c, cp}\tabularnewline
{\small op-sub-nudge} & {\small\texttt{0 mu}} & {\small\phantom{{\small f, fp, s, sp,}} c, cp}\tabularnewline
{\small{*}derivand-sep} & {\small\texttt{derivand-sep}} & {\small f, fp, s, sp, c, cp}\tabularnewline
{\small{*}op-set-left} & {\small\texttt{false}} & {\small f, fp, \phantom{{\small s, sp, c, cp,}} j}\tabularnewline
{\small{*}italic-nudge} & {\small\texttt{0 mu}} & {\small f, fp, \phantom{{\small s, sp, c, cp,}} j}\tabularnewline
{\small{*}inner-wrap} & {\small\texttt{false}} & {\small\phantom{{\small f, fp,}} s, sp}\tabularnewline
{\small{*}inner-Ldelim} & {\small\texttt{(}} & {\small\phantom{{\small f, fp,}} s, sp}\tabularnewline
{\small{*}inner-Rdelim} & {\small\texttt{)}} & {\small\phantom{{\small f, fp,}} s, sp}\tabularnewline
{\small{*}outer-Ldelim} & {\small\texttt{\textbackslash big {[}}} & {\small\phantom{{\small f, fp,}} s, sp}\tabularnewline
{\small{*}outer-Rdelim} & {\small\texttt{\textbackslash big {]}}} & {\small\phantom{{\small f, fp,}} s, sp}\tabularnewline
{\small{*}sub-nudge} & {\small\texttt{0 mu}} & {\small\phantom{{\small f, fp,}} s, sp}\tabularnewline
\bottomrule
\end{tabular}
\par\end{center}
\end{table}

\begin{description}
\item [{style}] the fraction form of derivative; 
\begin{itemize}
\item for upright-fraction derivatives, \verb`\diff`, \verb`\diffp`, a
choice of \verb`frac`, \verb`tfrac` or \verb`dfrac`:
\begin{itemize}
\item \verb`frac` results in a fraction formed from \verb`\frac`, scalable
\item \verb`tfrac` results in a fraction formed from \verb`\tfrac`, not
scalable
\item \verb`dfrac` results in a fraction formed from \verb`\dfrac`, not
scalable
\item default in templates \verb`DIFF`, \verb`DIFFP` = \verb`frac`
\end{itemize}
\item for slash-fraction derivatives,\verb`\difs`, \verb`\difsp`, a choice
of \verb`/`, \verb`auto`, \verb`big`, \verb`Big`, \verb`bigg`
or \verb`Bigg` 
\begin{itemize}
\item \verb`/` forms the slash fraction with \verb`/`, not scalable
\item \verb`auto` forms the slash fraction with \verb`\left. \middle/ \right.`,
scalable
\item \verb`big`, \verb`Big`, \verb`bigg` and \verb`Bigg` form the slash
fraction with \verb`\big/`, \verb`\Big/`, \verb`\bigg/` and \verb`\Bigg/`
respectively, not scalable
\item default in templates \verb`DIFS`, \verb`DIFSP` = \verb`/`
\end{itemize}
\item for compact-form derivatives, \verb`\difc`, \verb`\difcp`,  a choice
of \texttt{\_} or \verb`dl`
\begin{itemize}
\item \_ forms derivatives of compact form like $\difc yx$, $\difcp[1,2]{z}{x,y}$ 
\item \verb`dl` forms differentials like $\difc.dl.{}x$ and $\difcp.dl.[2]{}{x,y,z}$ 
\item default in templates \verb`DIFC`, \verb`DIFCP` = \texttt{\_}
\end{itemize}
\item overall default in template \verb`DIF` = \verb`frac`
\end{itemize}
\item [{slash-tok}] token or tokens used for the slash fraction; (see §\ref{subsec:Other-notations}
for a different assignment) default \verb`/`
\item [{slash-sep}] space inserted on either side of the \textbf{slash-tok};
 default \verb`0 mu`
\item [{derivand-sep}] horizontal space added before the differentiand
if the \verb`spaced` package option is set to \verb`1`, or before
a multi-tokened differentiand if the \verb`spaced` package option
is set to \verb`-1`; note that compact-form derivatives \emph{always
}have this space inserted; default (appropriate for an upright-fraction
derivative) = \verb`3mu plus 1mu minus 2mu`
\item [{op-symbol}] the operator symbol; for ordinary derivatives generally
one of \verb`d` or \verb`\mathrm{d}`, for partial derivatives \verb`\partial`;
default = \verb`\mathrm{d}`
\item [{op-symbol-alt}] if different from \textbf{op-symbol} then used
in the denominator of a fraction-form derivative while \verb`op-symbol`
is used in the numerator; e.g. for the acceleration $\diff.n.{v^{i}}t$,
\verb`op-symbol = \nabla` and \verb`op-symbol-alt = d`; defaults
to \textbf{op-symbol} default
\item [{op-order-nudge}] extra horizontal space added between the op-symbol
and the superscripted order of differentiation in higher order derivatives;
for  math-italic forms compare $d^{2}$ with $d^{\mkern1mu 2}$, $\partial^{2}$
with $\partial^{\mkern1mu 2}$ where the first symbol in each case
has no extra space and the second has an extra \verb`1 mu`; since
\textbf{op-symbol }defaults to an upright `d', default = \verb`0 mu`
\item [{var-sup-nudge}] extra horizontal space added between a variable
in the denominator of a derivative and the superscripted order of
differentiation in higher order derivatives (cf \textbf{op-order-nudge});
default = \verb`1 mu`
\item [{multi-term-sep}] horizontal spacing inserted between the differentials
in, for example, the denominator of a mixed partial derivative to
avoid a solid cluster like $\partial x\partial y\partial z$; with
the default \verb`2 mu` this is spread a little, $\difcp.dl.{}{x,y,z}$;
default = \verb`2 mu plus 1 mu minus 1 mu`
\item [{term-sep-adjust}] adjustment (usually a reduction) to \textbf{multi-term-sep}
when differentiation in a variable occurs to an order other than 1;
if, e.g., $\difcp.dl.[2]{}{x,y,z}$ is the denominator of a mixed
partial derivative in three variables, because of the superscript
the spacing between $\partial x^{2}$ and $\partial y$ is reduced
by\textbf{ term-sep-adjust} from the spacing between $\partial y$
and $\partial z$; default = {\ttfamily\verb`-1 mu`}
\item [{long-var-wrap}] to avoid ambiguity in higher order derivatives
it may aid clarity to wrap multi-token variables of differentiation
in parentheses; the choices are 
\begin{description}
\item [{\texttt{dv\phantom{()}}}] no wrapping, e.g. $\mathrm{d}x_{i}^{2}$
or $\partial\frac{1}{\Theta}^{2}$, 
\item [{\texttt{d(v)}}] wrap the variable only, e.g. $\mathrm{d}(x_{i})^{2}$
or  $\partial(\frac{1}{\Theta})^{2}$,
\item [{\texttt{(dv)}}] wrap both op-symbol and variable, e.g. $(\mathrm{d}x_{i})^{2}$
or $(\partial\frac{1}{\Theta})^{2}$;
\item [{\texttt{\phantom{(dv)}}}] default = \verb`d(v)`
\end{description}
\item [{lvwrap-Ldelim}] left delimiter when wrapping a long variable in
a higher order derivative; also applies to the left delimiter used
in a jacobian; default = \verb`\mleft (`
\item [{lvwrap-Rdelim}] right delimiter when wrapping a long variable in
a higher order derivative; also applies to the right delimiter used
in a jacobian; default = \verb`\mright )`
\item [{lvwrap-sup-nudge}] horizontal adjustment to the superscript position
when a multi-token variable is wrapped in (e.g.) parentheses and its
order of differentiation is not \verb`1`; default = \verb`-2 mu`
\item [{outer-Ldelim}] the left member of a delimiter pair wrapping the
derivative, the right member of which is subscripted to indicate a
point of evaluation or variables held constant; ISO recommends parentheses
for this purpose, hence default = {\ttfamily\verb`\left (`} 
\item [{outer-Rdelim}] the right member of a delimiter pair wrapping the
derivative and subscripted to indicate a point of evaluation or variables
held constant; ISO recommends parentheses for this purpose, hence
default = {\ttfamily\verb`\right )`}
\item [{elbowroom}] adjustment to the whitespace between \textbf{outer-Ldelim},
\textbf{outer-Rdelim }and the enclosed derivative; negative values
reduce the space; default = {\ttfamily\verb`0 mu`}
\item [{sub-nudge}] horizontal adjustment of the subscript's placing relative
to the \textbf{outer-Rdelim}iter for a point of evaluation or variable
held constant; a negative value compensates for the curving inwards
of a large right parenthesis; default = {\ttfamily\verb`-5 mu`}
\item [{op-sub-nudge}] horizontal adjustment of the position of the subscript
in derivatives of compact form relative to the operator; since \verb`\mathrm{d}`
is the default operator, default = \verb`0 mu`
\item [{{*}derivand-sep}] when the derivand is appended, horizontal space
added before the differentiand (derivand) depending on the setting
of the \verb`spaced` package option; default = \textbf{derivand-sep}
default
\item [{{*}op-set-left}] a choice of {\ttfamily\verb`true`}
or {\ttfamily\verb`false`} indicating whether the op-symbol
is left-aligned or not when the differentiand is appended; generally
it is centred; applies only to upright-fraction forms of the derivative;
default = {\ttfamily\verb`false`}
\item [{{*}italic-nudge}] if \textbf{{*}op-set-left} is \verb`true`, makes
an italic adjustment in the numerator, so that the op-symbols in numerator
and denominator align in the same slanting column; for \verb`d` or
\verb`\partial` an appropriate value might be \verb`3 mu`; because
of the default \verb`\mathrm{d}`, default = \verb`0 mu`
\item [{{*}inner-wrap}] when the differentiand is appended, a choice of
\verb`true` or \verb`false` dictating whether the differential operator
is wrapped in parentheses, as here $\difsp*{F(x,y)}x$, or not; for
a slash-fraction derivative \verb`true` is an appropriate default,
but the overall default, appropriate for an upright-fraction derivative,
= \verb`false`
\item [{{*}inner-Ldelim}] if \textbf{{*}inner-wrap} is \verb`true`, the\textbf{
}left member of a delimiter pair around the differential operator;
default = \verb`(`
\item [{{*}inner-Rdelim}] if \textbf{{*}inner-wrap} is \verb`true`, the\textbf{
}right member of a delimiter pair around the differential operator
; default = \verb`)`
\item [{{*}outer-Ldelim}] if \textbf{{*}inner-wrap} is \verb`true`, the
left member of a delimiter pair around both the differential operator
and appended differentiand, the right member of which may be subscripted
to indicate a point of evaluation or variables held constant; to avoid
too many parentheses, given the default values of {*}\textbf{inner-Ldelim},
{*}\textbf{inner-Rdelim}, default = \verb`\bigl [`
\item [{{*}outer-Rdelim}] if \textbf{{*}inner-wrap} is \verb`true`, the
right member of a delimiter pair around the differential operator
and appended differentiand; may be  subscripted to indicate a point
of evaluation or variables held constant; to avoid too many parentheses,
given the default values of \textbf{{*}inner-Ldelim}, \textbf{{*}inner-Rdelim},
default = \verb`\bigr ]`
\item [{{*}sub-nudge}] if \textbf{{*}inner-wrap} is \verb`true`, horizontal
adjustment of the subscript's placing relative to the \textbf{{*}outer-Rdelim}iter
for a point of evaluation or variable held constant; a negative value
compensates for the curving inwards of a large right parenthesis;
since the default \textbf{{*}outer-Rdelim} is a square bracket,\textbf{
}default = \verb`0 mu`
\end{description}

\subsection{Ordinary upright-fraction derivatives; template \texttt{DIFF}}

The defaults assigned in template \verb`DIF` are inherited by template
\verb`DIFF` without change. Template \verb`DIFF` is therefore strictly
unnecessary but, with templates \verb`DIFS` and \verb`DIFC` in mind,
was created for the sake of a consistent naming scheme. 

The \verb`\diff` command uses the values in the \verb`DIFF` template
to form an upright-fraction derivative. Only keys with an `f' in
the third column of Table~\ref{tab:DIF-defaults} are used in this
process. Keys without an `f' play no part in the process and their
default values are ignored. See §\ref{subsec:DIFF-(upright-fraction-derivative)}
for the complete list of \emph{relevant} \verb`DIFF` defaults.

\begin{table}
\caption{Defaults differing from the parent template}

\begin{centering}
\subfloat[\texttt{DIFS}\label{tab:DIFS-defaults}]{\centering{}%
\begin{tabular}{lr}
\toprule 
{\small key} & {\small default}\tabularnewline
\midrule
{\small style} & {\small\texttt{/}}\tabularnewline
{\small derivand-sep} & {\small\texttt{2muplus1muminus2mu}}\tabularnewline
{\small outer-Ldelim} & {\small\texttt{(}}\tabularnewline
{\small outer-Rdelim} & {\small\texttt{)}}\tabularnewline
{\small sub-nudge} & {\small\texttt{0 mu}}\tabularnewline
{\small{*}inner-wrap} & {\small\texttt{true}}\tabularnewline
\bottomrule
\end{tabular}}\hfil\subfloat[\texttt{DIFC}\label{tab:DIFC-defaults}]{\centering{}%
\begin{tabular}{lr}
\toprule 
{\small key} & {\small default}\tabularnewline
\midrule
{\small style} & {\small\texttt{\_}}\tabularnewline
{\small derivand-sep} & {\small\texttt{1muplus1muminus1mu}}\tabularnewline
{\small multi-term-sep} & {\small\texttt{1 mu}}\tabularnewline
{\small term-sep-adjust} & {\small\texttt{0 mu}}\tabularnewline
{\small outer-Ldelim} & {\small\texttt{\textbackslash bigl (}}\tabularnewline
{\small outer-Rdelim} & {\small\texttt{\textbackslash bigr )}}\tabularnewline
sub-nudge & {\small\texttt{-2 mu}}\tabularnewline
\bottomrule
\end{tabular}}
\par\end{centering}
\centering{}\subfloat[\texttt{DIFFP}\label{tab:DIFFP-defaults}]{\centering{}%
\begin{tabular}{lr}
\toprule 
{\small key} & {\small default}\tabularnewline
\midrule 
{\small op-symbol} & {\small\texttt{\textbackslash partial}}\tabularnewline
{\small op-order-nudge} & {\small\texttt{1 mu}}\tabularnewline
{\small{*}italic-nudge} & {\small\texttt{3 mu}}\tabularnewline
\bottomrule
\end{tabular}}\hfil\subfloat[\texttt{DIFSP, DIFCP}\label{tab:DIFSP-DIFCP-defaults}]{\centering{}%
\begin{tabular}{lr}
\toprule 
{\small key} & {\small default}\tabularnewline
\midrule 
{\small op-symbol} & {\small\texttt{\textbackslash partial}}\tabularnewline
{\small op-order-nudge} & {\small\texttt{1 mu}}\tabularnewline
\bottomrule
\end{tabular}}
\end{table}


\subsection{Ordinary slash-fraction derivatives; template \texttt{DIFS}}

When you use the command \verb`\difs` to form a slash-fraction derivative
it is the keys in template \verb`DIF` with an `s' in the third
column of Table~\ref{tab:DIF-defaults} which are used. Table~\ref{tab:DIFS-defaults}
records those keys used for this purpose which are assigned default
values \emph{different} from those in \verb`DIF`. See §\ref{subsec:DIFS-(slash-fraction-derivative)}
for the complete list of \emph{relevant} \verb`DIFS` defaults.

\subsection{Ordinary compact-form derivatives; template \texttt{DIFC}}

When you use the command \verb`\difc` to form a compact derivative
it is the keys in template \verb`DIF` with a `c' in the third column
of Table~\ref{tab:DIF-defaults} which are used. Table~\ref{tab:DIFC-defaults}
records those keys used for this purpose which are assigned default
values \emph{different} from those in \verb`DIF`. See §\ref{subsec:DIFC-(compact-derivative)}
for the complete list of \emph{relevant} \verb`DIFC` defaults.

\subsection{Partial derivatives; templates \texttt{DIFFP}, \texttt{DIFSP}, \texttt{DIFCP}}

The default values given in the tables so far apply to ordinary derivatives.
For \emph{partial} derivatives, only a few defaults change. These
are listed in Tables~\ref{tab:DIFFP-defaults}, \ref{tab:DIFSP-DIFCP-defaults}.
All other keys take the default values of the respective parent templates,
\verb`DIFF`, \verb`DIFS` and \verb`DIFC`. 

\section{Variant forms: the \texttt{\textbackslash difdef} command}

\label{sec:Variant-forms}You may be dissatisfied with the scheme
of default values listed in the preceding tables and wish to `Re-mould
it nearer to the Heart's Desire'. How to do so is discussed in §\ref{sec:ChangingDefaults}
below. In \emph{this} section it is assumed that the user is largely
satisfied with the assigned defaults but has need to write an occasional
derivative that deviates from the default form. For instance, to write
the range of different examples displayed in the Rogues' Gallery (§\ref{sec:Rogues'-gallery})
I had to make extensive use of such \emph{variant forms} of derivative.
I needed forms that displayed different ways of indicating a point
of evaluation, a form that showed a math-italic `d' rather than
the default upright `d', forms that displayed different parenthesizing
styles for higher-order derivatives with multi-token variable names,
and so on.

The process of defining and using such variants is a two-step process.
The `using' part is easy: you simply put the name of the variant
form between dots and append to the relevant \verb`\difx` or \verb`\difxp`
command.

The `defining' part makes use of a command \verb`\difdef`,
\begin{verbatim}
    \difdef{id-list}{variant-name}{key-value list}
\end{verbatim}
which has three \emph{mandatory} arguments: 
\begin{enumerate}
\item \verb`id-list` A comma-list of identifiers, one or more of \verb`f`,
\verb`s`, \verb`c`, \verb`fp`, \verb`sp`, \verb`cp`, \verb`j`,
\verb`l` distinguishing the respective templates \verb`DIFF`, \verb`DIFS`,
\verb`DIFC`, \verb`DIFFP`, \verb`DIFSP`, \verb`DIFCP`, \verb`DIFJ`
and \verb`DIFL` (for the last two see Chapter~\ref{chap:Differentials-and-jacobians}).
\item \verb`variant-name` A (preferably brief) name for the variant form;
it may include characters other than letters, like numbers, punctuation
marks (excluding full stops), mathematical symbols like \verb`+`
and \verb`=`, but not control sequences or active characters, nor
\verb`%` , \verb`#` or braces.
\item \verb`key-value list` A \emph{key=value} list where the settings
differ from the default settings for the relevant template or templates
(as determined by the \verb`id-list`).
\end{enumerate}
In the preamble to the present document I have included the following
definition:
\begin{verbatim}
    \difdef { f } { p } 
      {    
        op-symbol      = \partial,
        op-order-nudge = 1 mu
      }
\end{verbatim}
This defines a variant, with name \verb`p`, of an ordinary upright-fraction
derivative (the \verb`f` in the first argument) that displays as
a partial derivative. To use the variant simply append the name, as
a dot-delimited argument, to the \verb`\diff` command. For instance,
repeating an earlier example from thermodynamics,
\begin{centred}
\verb`\[ \diff.p.*{\frac PT}U[V] = \diff.p.*{\frac 1T}V[U] \]` $\Longrightarrow$
\[ \diff.p.*{\frac PT}U[V] = \diff.p.*{\frac 1T}V[U]\]
\end{centred}
\noindent The effect is exactly the same as previously and it would
have been possible to define \verb`\diffp` as this variant by following
the definition of the variant with the statement,
\begin{verbatim}
    \NewDocumentCommand \diffp { } { \diff.p. }
\end{verbatim}
\verb`diffcoeff` has not followed this path, instead choosing to
put the status and configurability of partial derivatives on the same
footing as ordinary derivatives.

\noindent %
\noindent\begin{minipage}[t]{1\columnwidth}%
\begin{shaded}%
The command \verb`\difdef` in version 5 of \verb`diffcoeff` takes
\emph{three} mandatory arguments for defining variant forms of derivative.
Do not confuse with the command \verb`\diffdef` of earlier versions
of \verb`diffcoeff` which took \emph{two }mandatory arguments for
this purpose. The third argument is required to identify which one
or more of the fraction forms \verb`f`, \verb`s`, \verb`c`, \verb`fp`,
\verb`sp`, \verb`cp` of the commands \verb`\difx`, \verb`\difxp`,
the variant applies to. In earlier versions this was not necessary
since there was only the one primary derivative command \verb`\diff`.\end{shaded}%
\end{minipage}

\medskip{}

The present document comes with a number of variant definitions. These
are divided into two groups. One, in the preamble, contains definitions,
like the example just given, designed to illustrate various effects
in this document -- as in the Rogues' Gallery. These preamble definitions
are listed in §\ref{sec:Preamble-definitions}. The other, in the
associated file \verb`diffcoeff5.def`, contains definitions that
may be of more general usefulness; these are listed in §\ref{sec:The-file-diffcoeff5.def}.

\subsection{The \texttt{.def} file}

\label{subsec:The-.def-file}A \verb`.def` file (in \verb`diffcoeff`)
is a text file containing a list of definitions of variant derivatives
after the fashion of the example above. The reason for placing such
variant definitions in a file is that they can be easily transferred
from document to document by means of the \verb`def-file` package
option. If the name of your \verb`.def` file is \verb`myfile`, then
invoking \verb`diffcoeff` with the call
\begin{verbatim}
    \usepackage[def-file=myfile]{diffcoeff}
\end{verbatim}
makes the definitions in \verb`myfile.def` available to your current
document -- provided \verb`diffcoeff` can find the file.

The question is: where to put the \verb`.def` file? The directory
of the current document is an obvious candidate and for the current
document serves well, but it does mean copying the \verb`.def` file
from directory to directory to work on \emph{different} documents.
To make a definition file available for \emph{all} documents, place
it in the \verb`texmf` tree, preferably not the one created by your
\TeX{} distribution, but your own \emph{personal} \verb`texmf` tree.
Provided your \TeX{} distribution knows about your personal \verb`texmf`
tree and the files it contains, then a \verb`.def` file placed within
it will be accessible to all documents.
\noindent \begin{center}
\noindent\begin{minipage}[t]{1\columnwidth}%
\begin{shaded}%

\subsubsection*{Personal \texttt{texmf} tree?}

This is a directory created by you for `waifs and strays' of the \TeX{}
system that are not included in standard distributions like MiK\TeX{}
or \TeX Live. For instance, it is the place for personal packages
designed for your own particular circumstances or preferences, and
is structured like the standard \texttt{texmf} hierarchy but placed
in another location so that there is no chance of its being overwritten
when your \TeX{} distribution is updated. But that distribution needs
to be alerted to the existence of your personal \texttt{texmf} tree
and any new files added to it. For MiK\TeX , open the MiK\TeX{} console,
click on \textsf{Settings} (in the column on the left) and then the
\textsf{Directories} tab. Click the \textsf{+} button and navigate
to your personal texmf tree to add it to the MiK\TeX{} search path,
using the arrow keys to place it as high in the search path as possible.
Having added it, you will then need to refresh the filename database
by clicking on the \textsf{Tasks} menu and selecting the obvious entry.
I am not familiar with \TeX Live but presume an analogous process
will apply there.\end{shaded}%
\end{minipage}
\par\end{center}

\begin{center}
\medskip{}
\noindent\begin{minipage}[t]{1\columnwidth}%
\begin{shaded}%

\subsubsection*{\texttt{diffcoeff.def}}

In earlier versions of \verb`diffcoeff`, if there was no explicit
\verb`def-file=<filename>` package option statement, then a file
\verb`diffcoeff.def` was searched for and if found loaded. This is
no longer the case. Version 5 of file of \verb`diffcoeff` searches
for a \verb`.def` \emph{only if it is explicitly named} in a package
option statement. (This decision was made at least in part to avoid
conflict with a \verb`diffcoeff.def` file from an earlier version
of \verb`diffcoeff` tucked away in some non-obvious place and producing
obscure errors in the current version 5.) \end{shaded}%
\end{minipage}
\par\end{center}

\subsubsection{Log file message}

If the\verb`.def` file named in the package option statement cannot
be located by \TeX , a message to that effect is sent to the terminal
and log file, but \verb`diffcoeff` continues loading.

\subsection{Examples of variants}

The dot-delimited name argument must always be the \emph{first} argument
of the \verb`\difx` or \verb`\difxp` command, even preceding an
asterisk (star) signalling `append the differentiand'. Now for some
examples.

\paragraph{Acceleration}

In tensor calculus acceleration is sometimes written $\difs.n.{v^{i}}t$,
where different operator symbols occur in numerator and denominator.
In the preamble to this document I have included the definition
\begin{verbatim}
    \difdef { f, s } { n }
      { 
        op-symbol     = \nabla,
        op-symbol-alt = \mathrm{d}
      }
\end{verbatim}
to give both upright- (the \verb`f`) and slash-fraction (the \verb`s`)
forms of the acceleration. Appending tne dot-delimited name \verb`n`
to \verb`\difs`, \verb`$ \difs.n.{v^i}t $` $\Longrightarrow$ $ \difs.n.{v^i}t $,
and appending the dot-delimited name \verb`n` to \verb`\diff`,
\begin{centred}
\verb`\[ \diff.n.{v^i}t. \]` $\Longrightarrow$ \[ \diff.n.{v^i}t. \]
\end{centred}

\paragraph{Detached subscripts}

To show the effect of the key \verb`sub-nudge`, the preamble contains
the definition
\begin{verbatim}
    \difdef { fp } { wsp } 
      { sub-nudge = 0 mu }
\end{verbatim}
The name \verb`wsp` is a contraction of `whitespace'. The defintion
applies only to an upright-fraction form of partial derivative (the
\verb`fp` in the optional argument). By giving \verb`sub-nudge`
a zero value in the \verb`wsp` variant, the subscript is cast adrift
(perhaps to float away?) on a sea of whitespace. The default setting,
\verb`sub-nudge=-5 mu`, maintains visual connection between subscript
and right parenthesis:

\verb`\[ \diffp.wsp.Fx[0],\quad \diffp Fx[0] \]` $\Longrightarrow$
\[ \diffp.wsp.Fx[0],\quad \diffp Fx[0] \]

\paragraph{Lagrange's equations}

An earlier example used Lagrange's equations of motion, which showed
a problem with the amount of whitespace introduced before a differentiand
bounded by a \verb`\left`, \verb`\right` pair. The file \verb`diffcoeff5.def`
contains the definition
\begin{verbatim}
    \difdef { f, fp } { *0 }
      { 
        *derivand-sep = 0 mu ,
        outer-Ldelim  = \mleft ( ,
        outer-Rdelim  = \mright )
      }
\end{verbatim}
The first thing to notice is that the name of the variant, \verb`*0`,
is not formed from letters (there are other examples below). Now Lagrange's
equations are rendered (just right to my eye!)
\begin{centred}
\verb`\[ \diffp L{q_{k}}-\diff.*0.**t{\diffp L{\dot{q}_{k}}[]}=0 \]`
\[
\diffp L{q_{k}}-\diff.*0.**t{\diffp L{\dot{q}_{k}}[]}=0.
\]
\end{centred}

\subsubsection{Editing variant forms}

\noindent %
\noindent\begin{minipage}[t]{1\columnwidth}%
\begin{shaded}%

\paragraph*{Bug in \texttt{xtemplate}}

If you wish to successfully \emph{edit} a variant form that has (already)
been defined -- by you or in \verb`diffcoeff` or in the \verb`.def`
file\emph{ }-- then you will need a version of \verb`xtemplate`
from 2022-12-17 or later. Earlier versions contained a bug that didn't
otherwise affect the workings of \verb`diffcoeff` but did prevent
changes being made to already defined variants.\end{shaded}%
\end{minipage}\medskip{}

You may wish to edit an already defined variant form -- perhaps to
give a \emph{negative} value to \verb`*derivand-sep` in the last
example. You don't need to repeat the full definition. It suffices
to change the setting only of the relevant key or keys:
\begin{verbatim}
    \difdef { f, fp } { *0 }
      { *derivand-sep = -3 mu }
\end{verbatim}
which gives for Lagrange's equations (too tight to my eye!)
\[
\difdef{f,fp}{*0}{*derivand-sep=-3mu}\diffp L{q_{k}}-\diff.*0.**t{\diffp L{\dot{q}_{k}}[]}=0.
\]


\subsubsection{Parenthesizing multi-token variables}

To illustrate the different modes of parenthesizing `long' variables
in higher order derivatives, I have put these two definitions in \verb`diffcoeff5.def`:
\begin{verbatim}
    \difdef { f, fp } { (dv) } 
      { long-var-wrap = (dv) } 

    \difdef { f, fp }{ dv } 
      { long-var-wrap = dv }
\end{verbatim}
The three possibilities for wrapping multitoken variables can now
be illustrated:
\begin{verbatim}
    \[ \diffp[2]f{x^{i}},\quad
       \diffp.dv.[2]f{x^{i}},\quad
       \diffp.(dv).[2]f{x^{i}} \] 
\end{verbatim}
$\Longrightarrow$ \[ \diffp[2]f{x^i},\quad\diffp.dv.[2]f{x^i},\quad\diffp.(dv).[2]f{x^i} \]

To my eye parenthesizing seems unnecessary in this case, but in the
following desirable (as in the first, default, setting):
\begin{verbatim}
    \[ \diffp[2]F{1/T},\quad
       \diffp.dv.[2]F{1/T},\quad
       \diffp.(dv).[2]F{1/T} \] 
\end{verbatim}
$\Longrightarrow$ \[ \diffp[2]F{1/T},\quad\diffp.dv.[2]F{1/T},\quad\diffp.(dv).[2]F{1/T} \]

\subsubsection{Point of evaluation}

\label{subsec:variantPoint-of-evaluation}Although ISO recommends
subscripting parentheses to indicate a point of evaluation, some (like
the author) prefer to subscript a vertical rule and save subscripted
parentheses for the case of variables held constant in partial derivatives.
The file \verb`diffcoeff5.def` contains the definition 
\begin{verbatim}
    \difdef { f, fp, s, sp } { | } 
      {
        outer-Ldelim  = \left . ,
        outer-Rdelim = \right |,
        sub-nudge   = 0 mu
      }
\end{verbatim}
where the `pipe' character is used for the name of the variant. 
\begin{centred}
\verb`\[ \diffp.|.{F(x,y)}x[x=1] \]` $\Longrightarrow$     \[ \diffp.|.{F(x,y)}x[x=1] \]
\end{centred}
For slash fractions, I think parentheses give a better result than
a vertical rule:
\begin{centred}
\verb`$ \difs yx[0],\quad \difs.|.yx[0] $` $\Longrightarrow$ $\difs yx[0],\quad \difs.|.yx[0]$
\end{centred}
They tie the whole expression together. However, it is easy to create
expressions that suffer from `parenthesis overload':
\begin{centred}
\verb`$ \difs{F(x)}{(2x)}[x=0],\quad\difs.|.{F(x)}{(2x)}[x=0] $`
$\Longrightarrow$ $\difs{F(x)}{(2x)}[x=0],\quad\difs.|.{F(x)}{(2x)}[x=0]$
\end{centred}
The vertical rule is better in this case, but best of all in this
case (to my eye) is the use of \emph{square} brackets. The file \verb`diffcoeff5.def`
contains the definition
\begin{verbatim}
    \difdef { f, fp, s, sp } { ] } 
      {
        outer-Ldelim = \left [ ,
        outer-Rdelim = \right ],
        elbowroom    = 1 mu,
        sub-nudge    = 0 mu
      }
\end{verbatim}
giving the result 
\begin{centred}
\verb`$ \difs.].{F(x)}{(2x)}[x=0] $` $\Longrightarrow$ $ \difs.].{F(x)}{(2x)}[x=0] $,
\end{centred}
which both avoids `parenthesis overload' and is `tied together'
by the square brackets (and at least gives a nod in the direction
of the ISO standard).

\subsubsection{Upright text-style derivatives}

\verb`diffcoeff` assumes that derivatives of upright-fraction form
will be used mainly in display-style expressions and that the slash
form will be used mainly for inline use. But if one does want to use
the fraction form in an inline expression, then \verb`$ \diffp ST $`
displaying as $ \diffp ST $ is fine, but adding a trailing optional
argument, \verb`$ \diffp ST[V] $`, to indicate (in the present example)
a variable held constant is not: $\diffp ST[V]$. Clearly the subscript
is too close to the right parenthesis and (to my eye) there is too
much `elbowroom' between the derivative and the enclosing parentheses.
Hence the file \verb`diffcoeff5.def` contains the following definition
for text-style upright fraction derivatives:
\begin{verbatim}
    \difdef { f, fp } { t }
      {
        style           = tfrac  ,
        derivand-sep    = 1 mu plus 1 mu minus 1 mu,
        multi-term-sep  = 0 mu   ,
        term-sep-adjust = 0 mu   ,
        wrap-sup-nudge  = 0 mu   ,
        outer-Ldelim    = \bigl (,
        outer-Rdelim    = \bigr ),
        elbowroom       = -2 mu  ,
        sub-nudge       = -3 mu
      }
\end{verbatim}
With this definition, the variant form \verb`$ \diffp.t.ST[V] $`
displays as $\diffp.t.ST[V]$. The subscript now is better positioned
and there is a better fit between parentheses and derivative. Note
that the \verb`style=tfrac` entry in the definition means \verb`\diffp.t.`
will not scale in a display-style environment and may give a ridiculous
result if used inappropriately:
\begin{centred}
\verb`\[ \frac AB\diffp.t.yx \]` $\Longrightarrow$ \[ \frac AB\diffp.t.yx \]
\end{centred}
For a non-scaling display-style derivative using \verb`\dfrac`, given
the defaults in templates \verb`DIFF` and \verb`DIFFP`, the definition
would be much simpler,
\begin{verbatim}
    \difdef { f, fp } { d } { style = dfrac }
\end{verbatim}
but there seems little point in so doing.

\subsubsection{Slash-fraction styles}

The default slash-fraction form \verb`$ \difs yx $` displaying as
$ \difs yx $ does not scale. It is intended for inline use, but sometimes
you may want a slash fraction of a different size -- perhaps a fraction
is present in the differentiand or in the variable of differentiation.
The file \verb`diffcoeff5.def` contains a definition of a scaling
slash fraction (name \verb`0`) and a slightly larger-than-default
slash fraction (name \verb`1`):
\begin{verbatim}
    \difdef { s, sp } { 0 }
      {
        style         = auto     ,
        outer-Ldelim  = \left [  ,
        outer-Rdelim  = \right ] ,
        sub-nudge     = 0 mu     ,
        *inner-Ldelim = \mleft ( ,
        *inner-Rdelim = \mright ),
        *outer-Ldelim = \left [  ,
        *outer-Rdelim = \right ]
      }
    \difdef { s, sp } { 1 }
      {
        style          =  big   ,
        outer-Ldelim   = \bigl (,
        outer-Rdelim   = \bigr ),
        sub-nudge      = -2.5 mu,
        *inner-Ldelim  = \bigl (,
        *inner-Rdelim  = \bigr ),
        *outer-Ldelim  = \bigl [,
        *outer-Rdelim  = \bigr ]
      }
\end{verbatim}
The names arise from the sequence \verb`\big/`, \verb`\Big/`, \verb`\bigg/`,
\verb`\Bigg/`, hence \verb`1`, \verb`2`, \verb`3`, \verb`4`,
which leaves \verb`0` for the scaling form (which is built around
\verb`\left.`, \verb`\middle/`, \verb`\right.`). \verb`diffcoeff5.def`
does not contain definitions for the \verb`2`, \verb`3`, \verb`4`
variants, only the two shown, because the larger sizes give ridiculous
results. For the scaling variant, it is also easy to produce eyesores:
\begin{centred}
\verb`\[ \difsp.0.{\frac1Y}{\frac1X} \]` $\Longrightarrow$ \[ \difsp.0.{\frac1Y}{\frac1X} \]
\end{centred}
But for small size increases, the results can be pleasing. To the
author's eye, both \verb`0` and \verb`1` variants give better results
than the default:
\begin{centred}
\verb`$ \difsp.1.{F(x,y)}{\tfrac1x}[0] $` $\Longrightarrow$ $ \difsp.1.{F(x,y)}{\tfrac1x}[0] $

\verb`$ \difsp.0.{F(x,y)}{\tfrac1x}[0] $` $\Longrightarrow$ $\difsp.0.{F(x,y)}{\tfrac1x}[0]\vphantom{\bigg|} $

\verb`$ \difsp{F(x,y)}{\tfrac1x}[0] $` $\Longrightarrow$ $ \difsp{F(x,y)}{\tfrac1x}[0] $
\end{centred}
Subscripted \emph{square} brackets are chosen for the scaling variant
so that the setting \verb`sub-nudge=0 mu` is appropriate at all scales.
They provide good visual contrast with the parentheses of $F(x,y)$.

\subsubsection{Compact--form derivatives}

Two styles are available for compact-form derivatives, derivative
style \verb`style=_`, and differential style \verb`style=dl`. The
first is the default style; the orders of differentiation are applied
to the operator symbol, in this example \verb`\partial`: 
\begin{centred}
\verb`$ \difcp[3,2]{f}{x,y,z} $` $\Longrightarrow$ $ \difcp[3,2]{f}{x,y,z} $
\end{centred}
But suppose we define a variant form (as is done in \verb`diffcoeff5.def`):
\begin{verbatim}
    \difdef { cp } { dl } { style = dl }
\end{verbatim}
and use it to form a similar expression but without the differentiand,
an empty argument in its place:
\begin{centred}
\verb`$ \difcp.dl.[3,2]{}{x,y,z} $` $\Longrightarrow$ $ \difcp.dl.[3,2]{}{x,y,z} $
\end{centred}
In this \verb`dl` style, the orders of differentiation are applied
to the \emph{variables}. This allows discussion of, for example, the
denominator of a mixed partial derivative -- perhaps a remark about
minutiae of spacing.

\subsubsection{D, \textbackslash delta, \textbackslash Delta derivatives}

\label{subsec:D-delta-Delta}In introductory calculus texts a derivative-like
symbol is created with the lowercase Greek delta, $\delta$ . An uppercase
Greek delta, $\Delta$, is often used in a derivative-like symbol
for an average. In fluid dynamics the \emph{material} (also \emph{substantive}
or \emph{total}\,) derivative uses an uppercase $\mathrm{D}$ in
place of $\mathrm{d}$. Texts on differential equations often use
a $\mathrm{D}$ operator. The file \verb`diffcoeff5.def` contains
the definitions
\begin{verbatim}
    \difdef { f, s } { gd }
      { op-symbol = \delta }
    \difdef { f, s } { gD }
      { op-symbol = \Delta }
    \difdef { f, s } { D }
      { op-symbol = \mathrm{D} }
    \difdef { c } { bD } 
      { 
        op-symbol    = \mathbf{D}, 
        op-sub-nudge = -2mu
      }
\end{verbatim}
(where the `g' in the first two suggests `greek'), meaning one
can write expressions like \verb`$ \difs.gd.yx $` $\Longrightarrow$
$ \difs.gd.yx $, or \verb`$ \difs.gD.yx $` $\Longrightarrow$ $ \difs.gD.st$
(for the average speed), or
\begin{centred}
\verb`\[ \diff.D.{\rho}t=\diffp\rho t + \mathbf{u\cdot}\nabla\rho \]`$\Longrightarrow$
\[ \diff.D.{\rho}t=\diffp\rho t + \mathbf{u\cdot}\nabla\rho \]
\end{centred}
\noindent for the total derivative of $\rho$ (perhaps in fluid dynamics),
or
\begin{centred}
\noindent \verb`$ \difc.bD.[2]y{x\,}+2\difc.bD.y{x\,}-4=0 $` $\Longrightarrow$
$ \difc.bD.[2]y{x\,}+2\difc.bD.y{x\,}-4=0 $
\end{centred}
for an example in the study of differential equations.

\subsection{Other notations}

\label{subsec:Other-notations}\verb`diffcoeff` and this document
are about defining \emph{derivatives} but it is worth pointing out
that other notations can be built from the \verb`diffcoeff` constituents,
in particular from the slash fraction forms. For example, some other
token than \verb`/`, or indeed series of tokens, can be used to link
numerator and denominator. It could be \verb`\vert` or \verb`\Vert`,
displaying as $\vert$ and $\Vert$ respectively, or \verb`\otimes`
(requiring for example \verb`\usepackage{stmaryrd}` in the preamble),
displaying as $\otimes$, or the sequence of tokens \verb`\otimes\ldots\otimes`
displaying as $\otimes\ldots\otimes$. The critical key is \verb`slash-tok`,
with possible extra spacing on either side through the key \verb`slash-sep`.
Or, one may want to void the \verb`op-symbol` key by giving it an
empty value or do something like \verb`op-symbol=\mathbf`, or give
\verb`outer-Ldelim`, \verb`outer-Rdelim` special values, e.g.,\verb`\langle`,
\verb`\rvert`.

In the preamble I have included the following definition, in order
to mimic the \verb`\Braket` command of the \verb`braket` package,
\begin{verbatim}
    \difdef{ s }{ bk }
      {
        slash-tok = ,
        op-symbol = ,
        multi-term-sep = 3mu\middle|\mskip3mu ,
        outer-Ldelim = \left\langle ,
        outer-Rdelim = \right\rangle
      }
\end{verbatim}
and supplemented it with the definition:
\begin{verbatim}
    \NewDocumentCommand \Braket { m }
      { 
        \difoverride {\negmu}
        \difs.bk.{}{#1}[] 
      }
\end{verbatim}
Testing the new command, \verb`\Braket`, gives this display:
\begin{centred}
\verb`\[ \Braket{\phi,\diffp[2]{}t,\psi} \]` $\Longrightarrow$ \[\Braket{\phi,\diffp[2]{}t,\psi} \]
\end{centred}
Comparison with the \verb`\Braket` command of the \verb`braket`
package, which uses \verb`|` as the separator in the argument rather
than commas, shows the displayed results to be the same (as far as
I can judge).

\section{Defaults: setting your own}

\label{sec:ChangingDefaults}The use of variant forms of derivative
assumes the user is reasonably satisfied with the default values of
the various templates. The user may not be. You may want different
defaults. That is again accomplished by means of the \verb`\difdef`
command. The procedure is identical with that for defining a variant
except that \emph{no} \verb`variant-name` is supplied; an empty argument
is used instead. 

For example, suppose you wish to indicate a point of evaluation for
ordinary upright-fraction derivatives by means of a subscripted vertical
rule rather than parentheses. In §\ref{subsec:variantPoint-of-evaluation}
we have seen how to create a variant form with this property but now
we want to make it the default in the template \verb`DIFF`. That
is easy -- simply omit any content from the second argument:
\begin{verbatim}
    \difdef { f } {} 
      {
        outer-Ldelim = \left . ,
        outer-Rdelim = \right |,
        sub-nudge    = 0 mu
      }
\end{verbatim}
\noindent If this is placed in the preamble of your document or in
your \verb`.def` file (see §\ref{subsec:The-.def-file}) then the
command  \verb`\diff yx[0]` will produce 
\[
\diff.|.yx[0]
\]
by default. By leaving the second argument empty the \verb`\difdef`
command has changed the default value of those templates indicated
by the list of identifiers in the first argument -- in the present
case only the template \verb`DIFF`. In particular, note that the
new default is \emph{not} inherited by \verb`DIFFP`. Inheritance
occurs \emph{only} at load time. How to change defaults that will
be inherited is discussed below in §\ref{subsec:Changing-DIF-defaults}.

If you wish to change other defaults of other templates follow the
same procedure. In the first argument of the \verb`\difdef` command
insert a comma-list of the derivative identifiers (\verb`f`, \verb`s`,
\verb`c`, \verb`fp`, \verb`sp`, \verb`cp`, \verb`j` or \verb`l`)
that you want the new defaults to apply to, \emph{leave the second
argument empty}, and in the third argument provide the \emph{key=value}
list of new defaults.

For instance, you might prefer math-italic `d's rather than the
upright ISO recommendation, and you want this to apply across \emph{all}
ordinary-derivative templates. The most straightforward way of achieving
that would be through the definition
\begin{verbatim}
    \difdef { f, s, c, l } {} 
      {
        op-symbol      = d ,
        op-order-nudge = 1 mu
      }
\end{verbatim}
which includes the list of identifiers \verb`f, s, c, l` in the first
argument (the \verb`l` referring to the template \verb`DIFL` of
the differential -- see §\ref{sec:Differentials}) and leaves the
second argument empty. Thereafter, all ordinary derivatives will be
graced with math-italic rather than upright `d's.

\subsection{Changing defaults in \texttt{DIF}}

\label{subsec:Changing-DIF-defaults}You might wonder if this last
effect could not have been obtained more simply by changing the default
in the `primogenitor' template \verb`DIF` -- perhaps leave both\emph{
first }and second\emph{ }arguments empty in the \verb`\difdef` command.
That, however, has no effect. The \verb`\difdef` command does nothing
if the first argument is empty. Inheritance occurs only once, at time
of birth -- \emph{load time} -- and not thereafter. If you want
to make a change affecting a number of templates by changing a default
in \verb`DIF`, then it has to be done at the time when \verb`diffcoeff`
is loaded. 

There are two ways to do this. The first is to create a text file
with the specific name \verb`diffcoeff.DIF` with the desired settings.
For example, if we want math-italic `d's and a subscripted vertical
rule for points of evaluation, then the file might look like
\begin{verbatim}
    op-symbol      = d,
    op-order-nudge = 1 mu,
    outer-Ldelim   = \left . ,
    outer-Rdelim   = \right |,
    sub-nudge      = 0 mu
\end{verbatim}
By locating the file in a place where your \TeX{} distribution can
find it -- either in the directory of the current document or in
your personal \verb`texmf` tree (see the earlier discussion at §\ref{subsec:The-.def-file},
and in particular the need to alert your \TeX{} distro to the presence
of the file) -- \verb`diffcoeff.DIF` will be read at load time and
the new defaults not only incorporated into template \verb`DIF` but
inherited by all child and grandchild templates unless explicitly
countermanded (for example by \verb`op-symbol = \partial` and similar
statements in the definitions of those templates).

The second method is to use the package option \verb`DIF`. For instance
loading \verb`diffcoeff` with the call
\begin{verbatim}
    \usepackage
      [
        DIF = { op-symbol = d,
          op-order-nudge  = 1 mu,
          outer-Ldelim    = \left . ,
          outer-Rdelim    = \right |,
          sub-nudge       = 0 mu }
      ]{diffcoeff}
\end{verbatim}
will overwrite the built-in defaults with these new values, which
will be inherited by child (and grandchild) templates unless explicitly
countermanded. Notice that since \verb`DIF` is a comma list it requires
braces around the list of \emph{key=value} statements\emph{.} 

If both methods of changing the template \verb`DIF` are employed,
the order of use is, first, read and act on the file \verb`diffcoeff.DIF`,
then read and act on the package option \verb`DIF`. (In other words,
to avoid complicating the preamble, preferably use the file \verb`diffcoeff.DIF`;
use the package option \verb`DIF` only for fine-tuning -- perhaps
a setting specific to that particular document.)

\chapter{Differentials and jacobians}

\label{chap:Differentials-and-jacobians}In addition to the six derivative
commands, \verb`\difx` and \verb`\difxp`,  the \verb`diffcoeff`
package has two further commands, \verb`\dl` and \verb`\jacob`,
for writing differentials and jabobian determinants respectively.
These commands use the settings of the templates \verb`DIFL` and
\verb`DIFJ`, and both are correspondingly configurable by means of
the \verb`\difdef` command.

\section{Differentials}

\label{sec:Differentials}Forms like $\dl x$ occur not only as components
of derivatives but also in other contexts like the expression for
a total differential,
\[
\dl P=\diffp Px\dl x+\diffp Py\dl y+\diffp Pz\dl z,
\]
or in integrals, like $\int\sin x\dl x$, or multi-variable integrals
like 
\[
\iiintop_{-\infty}^{\infty}V(x,y,z)\dl{x,y,z}.
\]
They also occur in differential geometry and elsewhere in the form
of line elements like 
\begin{centred}
$\dl.+.{x,y,z}^2$ \quad{}and\quad{} $c^2\dl.-.{t,x,y,z}^2$.
\end{centred}
Surely we want the `d's in these expressions to correspond to their
form (upright or math italic) in derivatives? To this end, \verb`diffcoeff`
provides a command \verb`\dl` to write the `d' in a differential
in a manner consistent with the default form used in derivatives.
In the present document, the default form is upright and so
\begin{centred}
\verb`$ \dl x $` $\Longrightarrow$ $ \dl x. $
\end{centred}
\noindent To use the command before a multi-token variable of differentiation,
put the variable in braces:
\begin{centred}
\verb`$ \dl{\vec{x}},\quad \dl{\mathbf{x}} $` $\Longrightarrow$
$ \dl{\vec{x}},\quad \dl{\mathbf{x}} $.
\end{centred}
For the triple integral above, writing the differentials required
not three but just the \emph{one} command:
\begin{centred}
\verb`$ \dl{x,y,z} $` $\Longrightarrow$ $ \dl{x,y,z} $.
\end{centred}
To write the line elements I made use of a dot-delimited argument
producing a variant form of the differential (see below §\ref{subsec:Line-elements}):
\begin{centred}
\verb`$ \dl.+.{x,y,z}^2 $` $\Longrightarrow$ $ \dl.+.{x,y,z}^2 $,

\verb`$ c^2\dl.-.{t,x,y,z}^2 $` $\Longrightarrow$ $ c^2\dl.-.{t,x,y,z}^2 $.
\end{centred}

\subsection{Template \texttt{DIFL}}

\begin{wraptable}{o}{0.5\columnwidth}%

\begin{centering}
\caption{\texttt{DIFL} defaults\protect\label{tab:DIFL-defaults}}
\begin{center}
\begin{tabular}{lrr}
\toprule 
{\small key} & {\small default} & {\small comment}\tabularnewline
\midrule
{\small style} & {\small\texttt{dl}} & {\small locked}\tabularnewline
{\small outer-Ldelim} & {\small\texttt{\textbackslash ,}} & \tabularnewline
{\small outer-Rdelim} &  & \tabularnewline
\bottomrule
\end{tabular}
\par\end{center}
\par\end{centering}
\end{wraptable}%
The differential command \verb`\dl` gives access to a template \verb`DIFL`
which inherits the default values of the fundamental template \verb`DIF`
with the (few) changes shown in Table~\ref{tab:DIFL-defaults}. Note
that the \verb`style` key is fixed at the value \verb`dl`; it cannot
be changed. The \verb`outer-Ldelim` key inserts a small space before
the differential; the \verb`outer-Rdelim` key does nothing. For the
differential, both \verb`outer-Ldelim` and \verb`outer-Rdelim` are
\emph{always inserted}. This differs from the derivative for which
\verb`outer-Ldelim` and \verb`outer-Rdelim` are inserted only if
there is a trailing optional argument. It is as if the differential
command \verb`\dl` had a built-in empty trailing optional argument.

That so few of the \verb`DIF` defaults are changed in \verb`DIFL`
indicates that much of the machinery of derivative formation is irrelevant
for forming a differential. A list of \emph{relevant} keys for the
creation of differentials -- those that have some effect on the appearance
of the thing -- can be found at §\ref{subsec:DIFL-(differential)}.

\subsection{Syntax and options}

\label{subsec:DifferentialsSyntaxOptions}If all options are present
the differential command has the syntax
\begin{verbatim}
    \dl.name.[order-spec]{variable(s)}^{exponent}
\end{verbatim}
where the arguments have the following significance:
\begin{enumerate}
\item \verb`name` (optional) A dot-delimited name to distinguish a variant
form (non-default form) of differential; see §\ref{subsec:Variant-differentials}
below.
\item \verb`order-spec` (optional) The power or comma-list of powers to
which the differential or differentials will be raised. If all powers
are \verb`1` then no specification is needed; indeed, if fewer powers
are specified than there are variables, all `missing' powers are
assumed to be \verb`1`; see the discussion for mixed partial derivatives,
§\ref{subsec:Mixed-partial-derivatives}.
\item \verb`variable(s)` (mandatory) The variable or comma-list of variables
the differential operator applies to. \verb`\dl x`, \verb`\dl{\vec{x}}`,
\verb`\dl{x,y,z}` are all valid variable specifications, displaying
as $\dl x$, $\dl{\vec{x}}$ and $\dl{x,y,z}$ respectively.
\item \verb`exponent` (optional) An exponent to which all differentials
will be raised; overrides the \verb`order-spec` ; see §\ref{subsec:Line-elements}
for examples of use.
\end{enumerate}
Only the third argument is mandatory, although it may be empty. 

\subsection{Variant forms of differential}

\label{subsec:Variant-differentials}The first argument of the differential
command \verb`\dl` is the optional \verb`name` which is used --
like the corresponding argument in the derivative commands -- to
define \emph{variant forms}.

To create such variant forms, the \verb`\defdif` command is again
used, but with \verb`l` (lowercase \verb`L`) used as the identifier
in the first argument. For example, you may want a `partial' differential,
using \verb`\partial` in place of \verb`d`. It seems natural to
give this the name \verb`p`:
\begin{verbatim}
    \difdef { l } { p }
      { op-symbol = \partial }
\end{verbatim}
In fact just this definition can be found in the file \verb`diffcoeff5.def`,
so that
\begin{centred}
\verb`$ \dl.p.x $` $\Longrightarrow$ $\dl.p.x$
\end{centred}
which is seven keystrokes in all versus ten (space included) for \verb`\partial x`.
Defining \verb`\dlp` by writing
\begin{centred}
\verb`\NewDocumentCommand \dlp {} { \dl.p. }`.
\end{centred}
saves another keystroke. However, I doubt the few keystrokes saved
justify the trouble of defining such a variant. The real reason one
might do so is to ease the writing of expressions like $\dl.p.[3,2]{x,y,z}$
-- perhaps in a document like the present one to discuss the minutiae
of spacing in the denominators of mixed partial derivatives. 
\begin{centred}
\verb`$\dl.p.[3,2]{x,y,z}$` $\Longrightarrow$ $\dl.p.[3,2]{x,y,z}$
\end{centred}
As you can see from the example, just as for mixed partial derivatives,
if more than one variable is specified but the \verb`order-spec`
contains fewer than that number of entries, \verb`diffcoeff` assumes
the missing entries are \verb`1`. 

A second example of a variant form of differential is provided by
the definition
\begin{verbatim}
   \difdef { l } { b }
      { op-symbol = \mathrm{d}\mathbf }
\end{verbatim}
which can be found in the file \verb`diffcoeff5.def`. If you distinguish
vectors, say, by boldface type, then you can avoid writing \verb`\mathbf`
for differentials of vectors by using the variant form \verb`\dl.b.`:
\begin{centred}
\verb`$ \dl.b.x,\quad \dl.b.{x,y,z} $` $\Longrightarrow$ $ \dl.b.x ,\quad \dl.b.{x,y,z} $.
\end{centred}

\subsubsection{Line elements}

\label{subsec:Line-elements}Variant forms can be used to write line
elements of Pythagorean or Minkowskian form. The definition
\begin{verbatim}
    \difdef { l } { + }
      {
        multi-term-sep  = 0 mu +,
        term-sep-adjust = 0 mu  ,
        outer-Ldelim      =
      }
\end{verbatim}
which can be found in the file \verb`diffcoeff5.def`, inserts a \verb`+`
sign between terms in the variable specification. Notice that the
value assigned to the key \verb`multi-term-sep` begins with \verb`0 mu`.
A dimension here \emph{initially} is essential. Also note that the
thin space inserted by default before a differential by means of the
\verb`outer-Ldelim` setting is now removed. But the intriguing feature
of the definition is what follows the \verb`0 mu` in the \verb`multi-term-sep`
value: a \verb`+` sign. Applying this variant to \verb`{x,y,z}`
the result is $ \dl.+.{x,y,z} $, which may be mildly interesting
but definitely becomes so when we add an exponent to the variable
spec.: 
\begin{centred}
\verb`$ \dl.+.{x,y,z}^2 $` $\Longrightarrow$ $ \dl.+.{x,y,z}^2 $.
\end{centred}
The exponent acts as if an order specification \verb`[2,2,2]` had
been included. If an order specification \emph{is }included, whatever
the values listed, the trailing exponent overrides it.

Similarly, the file \verb`diffcoeff5.def` contains an identical definition
save that the plus sign is replaced by a minus. This enables the writing
of a Minkowski metric:
\begin{centred}
\verb`$ c^2\dl.-.{t,x,y,z}^2 $` $\Longrightarrow$ $ c^2\dl.-.{t,x,y,z}^2 $.
\end{centred}

\subsection{Changing defaults}

To change the \emph{default} values of the \verb`DIFL` template use
the \verb`\difdef` command but leave its second argument, the \verb`name`
argument, empty. For instance if you want slightly less space by default
before a differential than the thin space (\verb`\,` or \verb`3 mu`)
specified in the \verb`DIFL` template -- say you want \verb`2 mu`
-- then write
\begin{verbatim}
    \difdef { l } {} { outer-Ldelim = \twomu }
\end{verbatim}
and ensure that this is in your \verb`.def` file or in the preamble
of your document. If you want a rubber length, say \verb`3 mu plus 1 mu minus 2 mu`
(which can also be written more compactly as \verb`3muplus1muminus2mu`),
then write (notice the \verb`\mskip`)
\begin{verbatim}
    \difdef { l } {} 
      { outer-Ldelim = \mskip 3muplus1muminus2mu }
\end{verbatim}
The crucial point is to leave the second argument of \verb`\difdef`,
the \verb`variant name`, empty. That changes the \emph{default} values
in \verb`DIFL` of the keys listed in the third argument of \verb`\difdef`.

\subsection{Rationale}

But why bother with the differential command at all? It only seems
to complicate the simple typing of \verb`d` followed by \verb`x`.
Admittely typing \verb`\dl x` requires fewer keystrokes than typing
\verb`\mathrm{d}x` (or even \verb`\mathrm dx`), but there are other,
more substantive, reasons why one might prefer an explicit command.
\begin{enumerate}
\item \emph{Consistency} with the derivative.
\item \emph{Spacing} is inserted automatically before the differential,
and between differentials in (e.g.) multiple integrals.
\item \emph{Parsing integrals} for some other package or program is much
easier to do when looking for a concluding differential command \verb`\dl`
than when looking for \verb`d` or \verb`\mathrm{d}` or \verb`\mathnormal{d}`
(or whatever) followed by an arbitrary variable name.
\item \emph{Configurability}. There are values other than the defaults that
can be given to keys to give novel effects for variant forms of differential
-- see the examples \verb`\dl.b.`, \verb`\dl.+.` and \verb`\dl.-.`
above. 
\end{enumerate}

\section{Jacobians}

\label{subsec:Jacobians}\verb`diffcoeff` provides a command {\ttfamily\verb`\jacob`}
for writing jacobians -- not the determinant as such but the symbol
conventionally used to denote the determinant. For example
\begin{centred}
\verb`\[ \jacob{u,v,w}{x,y,z} \]` $\Longrightarrow$ \[ \jacob{u,v,w}{x,y,z} \]
\end{centred}
The comma lists can contain any number of variables, even one or none,
\[
\jacob uv,\quad\jacob{}{},
\]
nor need the numbers in numerator and denominator be equal. \verb`\jacob`
does \emph{not} check such things. It is perfectly possible to form
unbalanced objects like
\begin{centred}
\verb`\[ \jacob{u,v,w}{x,y},\quad\jacob{u,v}{x,y,z}. \]` $\Longrightarrow$
\[ \jacob{u,v,w}{x,y},\quad\jacob{u,v}{x,y,z}. \]
\end{centred}
Perhaps there are contexts where these are meaningful?

\subsection{Template \texttt{DIFJ}}

\begin{wraptable}{o}{0.45\columnwidth}%
\centering{}\caption{\texttt{DIFJ} defaults\protect\label{tab:DIFJ-defaults}}
\begin{center}
\begin{tabular}{lr}
\toprule 
{\small key} & {\small default}\tabularnewline
\midrule
{\small op-symbol} & {\small\texttt{\textbackslash partial}}\tabularnewline
{\small outer-Ldelim} & \tabularnewline
{\small outer-Rdelim} & \tabularnewline
\bottomrule
\end{tabular}
\par\end{center}\end{wraptable}%
Jacobians are configurable. Like  other commands of \verb`diffcoeff`,
\verb`\jacob` gives access to a template, in this case \verb`DIFJ`,
which is a child of the fundamental template \verb`DIF` and inherits
most of its default values with only a few changes as shown in Table~\ref{tab:DIFJ-defaults}.
Note that the keys \verb`outer-Ldelim` and \verb`outer-Rdelim` are
both empty and, as with the differential, are \emph{always inserted}
-- which is why they are empty by default.

The lack of entries in Table~\ref{tab:DIFJ-defaults} is because
many keys are irrelevant for forming jacobians -- it doesn't matter
what their default values are. For a list of \emph{relevant} keys
-- ones that have some effect on the appearance of a jacobian --
see §\ref{subsec:DIFJ-(jacobian)}.

\subsection{Syntax and variant forms}

The \verb`\jacob` command has only three arguments. The syntax is
simple: 
\begin{verbatim}
    \jacob.name. {numer} {denom}
\end{verbatim}
The arguments have the following significance:
\begin{enumerate}
\item \verb`name` (optional) The dot-delimited name of a variant form of
jacobian.
\item \verb`numer` (mandatory) A comma list of variables forming the numerator
of the jacobian.
\item \verb`denom` (mandatory) A comma list of variables forming the denominator
of the jacobian.
\end{enumerate}
The default form of jacobian is an upright fraction with \verb`\partial`
operators and parentheses around the variable lists in both numerator
and denominator. 

If you want a jacobian in, say, slash-fraction form then once again
the \verb`\difdef` command is used. The file \verb`diffcoeff5.def`
contains the definition
\begin{verbatim}
    \difdef { j } { s } { style = / }
\end{verbatim}
To access this style, use the name -- which is at your discretion
but here I have chosen \verb`s` (\verb`/` also suggests itself)
-- between dots after the \verb`\jacob` command:
\begin{centred}
\verb`$ \jacob.s.{u,v,w}{x,y,z} $` $\Longrightarrow$ $ \jacob.s.{u,v,w}{x,y,z}. $
\end{centred}
If you want to change the operator symbol from \verb`\partial` to
\verb`D`, as I have seen used, then the definition is:
\begin{verbatim}
    \difdef { j } { D } { op-symbol = D }
\end{verbatim}
(Again the name is at your discretion but \verb`D` seems obvious.)
I have added this to the preamble of the present document, so that
\begin{centred}
\verb`\[ \jacob.D.{u,v,w}{x,y,z} \]` $\Longrightarrow$ \[ \jacob.D.{u,v,w}{x,y,z}.\]
\end{centred}
If you want square brackets rather than parentheses around the variable
lists, then \verb`lvwrap-Ldelim` and \verb`lvwrap-Rdelim` (perhaps
not intuitively) are the keys to change: 
\begin{verbatim}
    \difdef { j } { [ } 
      { 
        lvwrap-Ldelim = \onemu\mleft [,
        lvwrap-Rdelim = \mright ] 
      }
\end{verbatim}
the \verb`\onemu` giving, to my eye, better spacing between the \verb`\partial`
symbols and the left brackets. This definition, too, has been added
to the preamble so that
\begin{centred}
\verb`\[ \jacob.[.{u,v,w}{x,y,z} \]` $\Longrightarrow$ \[ \jacob.[.{u,v,w}{x,y,z}. \]
\end{centred}

\subsection{Changing defaults}

To change the \emph{default} values of the \verb`DIFJ` template leave
the second argument of the \verb`\difdef` command -- the \verb`variant-name`
-- empty. For instance, if you want square brackets to be your default
setting, the \verb`\difdef` command would be 
\begin{verbatim}
    \difdef { j } {} 
      { 
        lvwrap-Ldelim = \onemu\mleft [,
        lvwrap-Rdelim = \mright ] 
      }
\end{verbatim}
The only difference from the previous definition is the absence of
the name from the second argument, which is now empty. If this definition
were added to the preamble or to the \verb`.def` file of your current
document then writing \verb`\jacob{u,v,w}{x,y,z}` would give the
same result as obtained above with the variant \verb`\jacob.[.{u,v,w}{x,y,z}`.

\chapter{Reference}

\label{chap:Reference}For convenience I list here the commands of
\verb`diffcoeff`, the template defaults, and the files and preamble
definitions associated with this document.

\section{Commands}

\verb`\diff`, \verb`\diffp`, \verb`\difs`, \verb`\difsp`, \verb`\difc`,
\verb`\difcp` (sometimes summarised as \verb`\difx` and \verb`\difxp`),
ordinary and partial derivatives of upright-fraction, slash-fraction
and compact forms respectively, with arguments (all optional unless
otherwise indicated) and their delimiters:
\begin{enumerate}
\item \verb`.name.` name of variant form of derivative;
\item \verb`*` append-differentiand switch;
\item \verb`*` reverse order of mandatory arguments 6 and 7 when differentiand
is appended, available only if first star is also present; 
\item \verb`[order(s)]` order of differentiation, or comma list of orders
of differentiation (for mixed partial derivatives);
\item \verb`{differentiand}` (mandatory) function being differentiated;
\item \verb`{variable(s)}` (mandatory) differentiation variable or, for
mixed partial derivatives, comma list of differentiation variables;
\item \verb`[pt of eval/const vars]` point of evaluation or, for partial
derivatives, variables held constant.
\end{enumerate}
\begin{description}
\item [{\texttt{\textbackslash difoverride}}] order-override command with
one mandatory argument:
\end{description}
\begin{enumerate}
\item \verb`{total order}` total order of differentiation; may be (and
generally is) empty.\newpage{}
\end{enumerate}
\begin{description}
\item [{\texttt{\textbackslash jacob}}] jacobian with arguments and delimiters:
\end{description}
\begin{enumerate}
\item \verb`.name.` (optional) name of variant form of jacobian;
\item \verb`{numer}` (mandatory) comma list of variables forming the numerator;
\item \verb`{denom}` (mandatory) comma list of variables forming the denominator.
\end{enumerate}
\begin{description}
\item [{\texttt{\textbackslash dl}}] differential with arguments and delimiters:
\begin{enumerate}
\item \verb`.name.` (optional) name of variant form of differential;
\item \verb`[order(s)]` (optional) order of differential or comma list
of orders of differentials;
\item \verb`{variable(s)}` (mandatory) variable or comma list of variables;
\item \verb`^{exponent}` (optional) exponent that overrides the \verb`order(s)`
specification, raising each differential to this power.
\end{enumerate}
\item [{\texttt{\textbackslash difdef}}] with arguments (all mandatory)
and delimiters: 
\begin{enumerate}
\item \verb`{id(s)}` comma list of one, some or all of the identifiers
\verb`f`, \verb`s`, \verb`c`, \verb`fp`, \verb`sp`, \verb`cp`,
\verb`j`, \verb`l` identifying upright fraction, slash fraction
and compact ordinary derivatives; upright fraction, slash fraction
and compact partial derivatives, and jacobians and differentials;
\item \verb`{name}` name for a variant form of derivative; as well as letters
may include numbers and other keyboard characters, but not braces,
\verb`%` or \verb`#`;
\item \verb`{settings}` comma list of changed \emph{key=value} settings.
\end{enumerate}
\item [{\texttt{\textbackslash negmu}}] insert a $-1$ mu space 
\item [{\texttt{\textbackslash nilmu}}] insert a $0$ mu space 
\item [{\texttt{\textbackslash onemu}}] insert a $1$ mu space 
\item [{\texttt{\textbackslash twomu}}] insert a $2$ mu space
\end{description}

\section{Templates}

The following lists record the default values of the templates used
by \verb`diffcoeff`. A marginal \verb`>` indicates where a setting
differs from that in \verb`DIF`, \verb`>>` where a setting differs
from that in \verb`DIFF`, \verb`DIFS` or \verb`DIFC` as the case
may be. For the latter templates, only \emph{relevant }keys have been
listed -- those which affect the appearance of the derivative (or
jacobian or differential).

\subsection{\texttt{DIF} (primogenitor)}
\begin{verbatim}
    style            = frac,
    slash-tok        = /,
    slash-sep        = 0 mu,
    derivand-sep     = 3 mu plus 1 mu minus 2 mu,
    op-symbol        = \mathrm{d},
    op-symbol-alt    = \KeyValue{ op-symbol },
    op-order-nudge   = 0 mu,
    var-sup-nudge    = 1 mu,
    multi-term-sep   = 2 mu plus 1 mu minus 1 mu,
    term-sep-adjust  = -1 mu,
    long-var-wrap    = d(v),
    lvwrap-Ldelim    = \mleft (,
    lvwrap-Rdelim    = \mright ),
    lvwrap-sup-nudge = -2 mu,
    outer-Ldelim     = \left (,
    outer-Rdelim     = \right ),
    elbowroom        = 0 mu,
    sub-nudge        = -5 mu,
    op-sub-nudge     = 0 mu,
    *derivand-sep    = \KeyValue{ derivand-sep },
    *op-set-left     = false,
    *italic-nudge    = 0 mu,
    *inner-wrap      = false,
    *inner-Ldelim    = (,
    *inner-Rdelim    = ),
    *outer-Ldelim    = \big [,
    *outer-Rdelim    = \big ],
    *sub-nudge       = 0 mu
\end{verbatim}

\subsection{\texttt{DIFF} (upright-fraction derivative)}

\label{subsec:DIFF-(upright-fraction-derivative)}Relevant keys and
default values for template \verb`DIFF`.
\begin{verbatim}
    style            = frac,
    derivand-sep     = 3 mu plus 1 mu minus 2 mu,
    op-symbol        = \mathrm{d},
    op-symbol-alt    = \KeyValue { op-symbol },
    op-order-nudge   = 0 mu,
    var-sup-nudge    = 1 mu,
    multi-term-sep   = 2 mu plus 1 mu minus 1 mu,
    term-sep-adjust  = -1 mu,
    long-var-wrap    = d(v),
    lvwrap-Ldelim    = \mleft (,
    lvwrap-Rdelim    = \mright ),
    lvwrap-sup-nudge = -2 mu,
    outer-Ldelim     = \left (,
    outer-Rdelim     = \right ),
    elbowroom        = 0 mu,
    sub-nudge        = -5 mu,
    *derivand-sep    = \KeyValue { derivand-sep },
    *op-set-left     = false,
    *italic-nudge    = 0 mu 
\end{verbatim}

\subsubsection{\texttt{DIFFP}}

\verb`DIFF` defaults as above with these changes:
\begin{verbatim}
>>  op-symbol        = \partial,
>>  op-order-nudge   = 1 mu,
>>  *italic-nudge    = 3 mu
\end{verbatim}

\subsection{\texttt{DIFS} (slash-fraction derivative)}

\label{subsec:DIFS-(slash-fraction-derivative)}Relevant keys and
default values for template \verb`DIFS`.
\begin{verbatim}
>   style            = /,
    slash-tok        = /,
    slash-sep        = 0 mu,
>   derivand-sep     = 2 mu plus 1 mu minus 2 mu,
    op-symbol        = \mathrm{d},
    op-symbol-alt    = \KeyValue { op-symbol },
    op-order-nudge   = 0 mu,
    var-sup-nudge    = 1 mu,
    multi-term-sep   = 2 mu plus 1 mu minus 1 mu,
    term-sep-adjust  = -1 mu,
    long-var-wrap    = d(v),
    lvwrap-Ldelim    = \mleft (,
    lvwrap-Rdelim    = \mright ),
    lvwrap-sup-nudge = -2 mu,
>   outer-Ldelim     = (,
>   outer-Rdelim     = ),
    elbowroom        = 0 mu,
>   sub-nudge        = 0 mu,
    *derivand-sep    = \KeyValue { derivand-sep },
>   *inner-wrap      = true, 
    *inner-Ldelim    = (,
    *inner-Rdelim    = ),
    *outer-Ldelim    = \big [,
    *outer-Rdelim    = \big ],
    *sub-nudge       = 0 mu
\end{verbatim}

\subsubsection{\texttt{DIFSP}}

\verb`DIFS` defaults as above with these changes:
\begin{verbatim}
>>  op-symbol        = \partial,
>>  op-order-nudge   = 1 mu 
\end{verbatim}

\subsection{\texttt{DIFC} (compact derivative)}

\label{subsec:DIFC-(compact-derivative)}Relevant keys and default
values for template \verb`DIFC`.
\begin{verbatim}
>   style            = _ ,
>   derivand-sep     = 1 mu plus 1 mu minus 2 mu,
    op-symbol        = \mathrm{d},
    op-order-nudge   = 0 mu,
>   multi-term-sep   = 1 mu,
>   term-sep-adjust  = 0 mu,
>   outer-Ldelim     = \bigl (,
>   outer-Rdelim     = \bigr ),
    elbowroom        = 0 mu,
>   sub-nudge        = -2 mu,
    op-sub-nudge     = 0 mu,
    *derivand-sep    = \KeyValue { derivand-sep }
\end{verbatim}

\subsubsection{\texttt{DIFCP}}

\verb`DIFC` defaults as above with these changes:
\begin{verbatim}
>>  op-symbol        = \partial,
>>  op-order-nudge   = 1 mu 
\end{verbatim}

\subsection{\texttt{DIFJ} (jacobian)}

\label{subsec:DIFJ-(jacobian)}Relevant keys and default values for
template \verb`DIFJ`.
\begin{verbatim}
    style            = frac,
    slash-tok        = /,
    slash-sep        = 0 mu,
>   op-symbol        = \partial,
    op-symbol-alt    = \KeyValue{ op-symbol },
    lvwrap-Ldelim    = \mleft (,
    lvwrap-Rdelim    = \mright ),
>   outer-Ldelim     = ,
>   outer-Rdelim     = ,
    elbowroom        = 0 mu ,
    *op-set-left     = false,
    *italic-nudge    = 0 mu 
\end{verbatim}

\subsection{\texttt{DIFL} (differential)}

\label{subsec:DIFL-(differential)}Relevant keys and default values
for template \verb`DIFL`.
\begin{verbatim}
    op-symbol        = \mathrm{d},
    var-sup-nudge    = 1 mu,
    multi-term-sep   = 2 mu plus 1 mu minus 1 mu,
    term-sep-adjust  = -1 mu,
>   long-var-wrap    = dv,
    lvwrap-Ldelim    = \mleft (,
    lvwrap-Rdelim    = \mright ),
    lvwrap-sup-nudge = -2 mu,
>   outer-Ldelim     = \, ,
>   outer-Rdelim     = ,
    elbowroom        = 0 mu
\end{verbatim}

\section{The file \texttt{diffcoeff5.def}}

\label{sec:The-file-diffcoeff5.def}
\begin{verbatim}
% file `diffcoeff5.def'
% definitions for variant forms
% 2023/01/03
% Andrew Parsloe ajparsloe@gmail.com
% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% material derivative
\difdef { f, s } { D }
  { op-symbol = \mathrm{D} }
% math italic
\difdef { f, s, c } { d' } 
  {
    op-symbol      = d,
    op-order-nudge = 1 mu 
  }
\difdef { f, s, c } { D' } 
  {
    op-symbol  = D,
    op-order-nudge = 1 mu
  }
% Greek 
\difdef { f, s } { gd }
  { op-symbol = \delta }
\difdef { f, s } { gD }
  { op-symbol = \Delta }
% spaceless appending
\difdef { f, fp } { *0 }
  { 
    *derivand-sep = 0 mu     ,
    outer-Ldelim  = \mleft ( ,
    outer-Rdelim  = \mright )
  }
% tfrac, nonscalable
\difdef { f, fp } { t }
  {
    style             = tfrac  ,
    derivand-sep      = 1 mu plus 1 mu minus 1 mu,
    multi-term-sep    = 0 mu   ,
    term-sep-adjust   = 0 mu   ,
    lvwrap-sup-nudge  = 0 mu   ,
    outer-Ldelim      = \bigl (,
    outer-Rdelim      = \bigr ),
    elbowroom         = -2 mu  ,
    sub-nudge         = -3 mu
  }
% slash fractions: 0=scalable,
% 1=big, 2=Big, 3=bigg, 4=Bigg
% but > 1 gives eyesores
\difdef { s, sp } { 0 }
  {
    style         = auto     ,
    outer-Ldelim  = \left [  ,
    outer-Rdelim  = \right ] ,
    sub-nudge     = 0 mu    ,
    *inner-Ldelim = \mleft ( ,
    *inner-Rdelim = \mright ),
    *outer-Ldelim = \left [  ,
    *outer-Rdelim = \right ]
  }
\difdef { s, sp } { 1 }
  {
    style          =  big   ,
    outer-Ldelim   = \bigl (,
    outer-Rdelim   = \bigr ),
    sub-nudge      = -2.5 mu,
    *inner-Ldelim  = \bigl (,
    *inner-Rdelim  = \bigr ),
    *outer-Ldelim  = \bigl [,
    *outer-Rdelim  = \bigr ]
  }
% vrule point of evaluation 
\difdef { f, fp, s, sp } { | } 
  {
    outer-Ldelim = \left . ,
    outer-Rdelim = \right |,
    sub-nudge    = 0 mu
  }
% sq. bracket pt of eval.
\difdef { f, fp, s, sp } { ] } 
  {
    outer-Ldelim = \left [ ,
    outer-Rdelim = \right ],
    elbowroom    = 1 mu,
    sub-nudge    = 0 mu
  }
% long var wrap
\difdef { f, fp } { (dv) } 
  { long-var-wrap = (dv) }
\difdef { f, fp } { dv } 
  { long-var-wrap = dv }
% compact, D operator
\difdef { c } { D }
  { 
    op-symbol    = \mathrm{D}, 
    op-sub-nudge = -2mu
  }
\difdef { c } { D' }
  { 
    op-symbol    = D, 
    op-sub-nudge = -2mu
  }
% bold
\difdef { c } { bD } 
  { 
    op-symbol    = \mathbf{D}, 
    op-sub-nudge = -2mu
  }
% differential style
\difdef { c, cp } { dl } 
  { style = dl }
%%%%%%%%%%% differential %%%%%%%%%%
% partial
\difdef { l } { p }
  { op-symbol = \partial }
% bold
\difdef { l } { b }
  { op-symbol = \mathrm{d}\mathbf }

% line elements: Pythagoras (+)
\difdef { l } { + }
  {
    multi-term-sep  = 0 mu +,
    term-sep-adjust = 0 mu  ,
    outer-Ldelim    =
  }
% Minkowski (-)
\difdef { l } { - }
  {
    multi-term-sep  = 0 mu -,
    term-sep-adjust = 0 mu  ,
    outer-Ldelim    =
  }
%%%%%%%%%% jacobian %%%%%%%%%%
% slash fraction
\difdef { j } { s }
  { style = / }
\end{verbatim}

\section{Preamble definitions}

\label{sec:Preamble-definitions}The preamble to the present document
contains the command
\begin{verbatim}
\usepackage[def-file=diffcoeff5,spaced=-1]{diffcoeff}
\end{verbatim}
and definitions:
\begin{verbatim}
% nabla in numer, d in denom
\difdef { f, s } { n } 
  {
    op-symbol     = \nabla,
    op-symbol-alt = \mathrm{d}
  }
% no sub nudge (a sea of white space)
\difdef { fp } { wsp } 
  { sub-nudge = 0 mu }
% align op left; no italic nudge 
\difdef { f } { left0 } 
  {
    *op-set-left  = true,
    *italic-nudge = 0 mu
  }
% align op left; italic nudge
\difdef { fp } { left } 
  {
    op-symbol      = \partial,
    op-order-nudge = 1 mu,
    *op-set-left  = true,
    *italic-nudge = 3 mu
  }
% partial variant of \diff
\difdef { f } { p }
  {
    op-symbol      = \partial,
    op-order-nudge = 1 mu
  }
% partial, 3mu sep of terms
\difdef { fp, sp } { 3mu } 
  {  multi-term-sep  = 3 mu }
% D jacobian
\difdef { j } { D } 
  { op-symbol = D }
% square bracket jabobian
\difdef { j } { [ } 
  { 
    lvwrap-Ldelim = \onemu\mleft [,
    lvwrap-Rdelim = \mright ] 
  }
% mimicking the \Braket command
% of the braket package
\difdef{ s }{ bk }
  { 
      slash-tok = ,
      op-symbol =  ,
      multi-term-sep = 3mu\middle|\mskip3mu ,
      outer-Ldelim=\left\langle ,
      outer-Rdelim=\right\rangle
  }
\NewDocumentCommand \Braket { m }
  { 
    \difoverride {\negmu}
    \difs.bk.{}{#1}[] 
  }
\end{verbatim}

\section{\texttt{\textbackslash DeclareChildTemplate}}

\verb`xtemplate` provides only a single function, \verb`\DeclareRestrictedTemplate`,
for creating a child template from a parent. \emph{All} the keys of
the child template are present in the parent. The child inherits not
only the keys of the parent but the default settings of those keys.
Some of those settings are `marked' so that they cannot be changed
by any \emph{instance} of the child (the \emph{restricted} keys) . 

Unfortunately there is no similar function available in \verb`xtemplate`
at present by which one can create a child with \emph{new} as well
as restricted default values. I found I could achieve this functionality
with the following code cobbled together from publicly declared functions
in \verb`xtemplate`. 
\begin{verbatim}
% Child template with both new and restricted defaults
% #1 object; #2 parent template; #3 child template; 
% #4 restricted, #5 new defaults (both key=value)
\NewDocumentCommand \DeclareChildTemplate { m m m m m }
  { 
    \DeclareRestrictedTemplate {#1} {#2} {#3} {}
    \EditTemplateDefaults {#1} {#3} {#5}
    \DeclareRestrictedTemplate {#1} {#3} {#3} {#4}
  }
\end{verbatim}
The first \verb`\DeclareRestrictedTemplate` call creates the child
template \verb`#3` from the parent template \verb`#2`, inheriting
all its keys and default values. \emph{No} restrictions are imposed
at this stage because the following \verb`\EditTemplateDefaults`
would immediately cancel them. That statement specifies the \emph{new}
defaults \verb`#5` of the child -- those that differ from the parent.
The default settings \verb`#4` of the parent that are restricted
to particular values in the child are imposed by the second \verb`\DeclareRestrictedTemplate`
call through the aritifice of treating the child template \verb`#3`
as a child of itself. In that way its new defaults are not lost.

\section{Version history}

Version 5 was conceived as a new package (under the name \verb`diffcoefx`)
and only at the end, after discussion with CTAN maintainers, changed
to version 5.0 of \verb`diffcoeff`. 
\begin{enumerate}
\item Version 5.0 (2023-01-02) of \verb`diffcoeff`
\begin{enumerate}
\item splits the \verb`\diff` command of version 4 into three pairs of
commands: \verb`\diff` and \verb`\diffp` for upright-fraction derivatives;
\verb`\difs` and \verb`difsp` for slash-fraction derivatives, and
\verb`\difc` and \verb`\difcp` for `compact form' derivatives;
\item replaces the order-override option by a new command \verb`\difoverride`
(to avoid cluttering formulas with a second square-bracket delimited
optional argument before the differentiand);
\item adds a second star option to reverse the order of differentiand and
variable(s) of differentiation when the differentiand is appended;
\item replaces the two-argument \verb`\diffdef` command of earlier versions
with the three-argument command \verb`\difdef` command, the additional
argument determining which one or more of the \verb`f`, \verb`s`,
\verb`c`, \verb`fp`, \verb`sp` or \verb`cp` forms the defined
variant applies to;
\item rewrites the differential command \verb`\dl` which is now template-configurable
(e.g. allowing easy writing of line elements like $\dl.+.{x,y,z}^{2}$);
\item rewrites the jacobian command \verb`\jacob` which is now template-configurable;
\item uses ISO defaults;
\item includes version conflict messages.
\end{enumerate}
\item Version 5.1 (2023-01-16)
\begin{enumerate}
\item adds a now-redundant \verb`ISO` package option and related version
conflict message;
\item makes some corresponding tweaks to documentation (including this version
5 history).
\end{enumerate}
\end{enumerate}

\end{document}
