Howto: if argument is numerical ... else

Question

I'm trying to create a differentiation command. At the moment I have the following:

\newcommand{\diff}[1][]{%
  \def\ArgI{{#1}}%
  \diffRelay%
}
\newcommand{\diffRelay}[2][]{%
  \frac{\textnormal{d}^{#1}\ArgI}{\textnormal{d}#2^{#1}}%
}

This allows me to use it like \diff{x}, diff[y]{x}, diff[y][2]{x} to create diff-by-x, diff-y-by-x, second-derivative-of-y-by-x. To simply write the n-th derivative of anything I have to use \diff[][n]{x}.

What I'm looking for is a way to check if the first optional argument is numerical, so I can omit the empty argument to create n-th derivative:

\newcommand{\diff}[1][]{%
  \def\ArgI{{#1}}%
  \diffRelay%
}
\newcommand{\diffRelay}[2][]{%
  \if \ArgI is numerical % better: \and #1 is empty
    \frac{\textnormal{d}^{\ArgI}}{\textnormal{d}#2^{\ArgI}}%
  \else
    \frac{\textnormal{d}^{#1}\ArgI}{\textnormal{d}#2^{#1}}%
}

Answer 1

I think you want something like this (which actually tests for a sequence of digits resulting in >0 so don't use [000] ;-)

\documentclass{article}

\newcommand{\diff}[1][]{%
  \def\ArgI{#1}%
  \diffRelay%
}
\def\eatrelax#1\relax{}
\newcommand{\diffRelay}[2][]{%
  \expandafter\eatrelax\ifnum0=0\ArgI\relax% is numerical % better: \and #1 is empty
    \frac{\textnormal{d}^{#1}\ArgI}{\textnormal{d}#2^{#1}}%
  \else\relax
    \frac{\textnormal{d}^{\ArgI}}{\textnormal{d}#2^{\ArgI}}%
 \fi
}

\begin{document}

$\diff[2]{x}$

$\diff[y]{x}$

$\diff[y][2]{x}$

\end{document}

Answer 2

You don't need the test for numeric, in my opinion. I suggest a different syntax:

\der[<function>]{<variable>}[<order>]

with two optional arguments; if you omit the first, you get the operator, otherwise the derivative; the second argument is the order of derivation.

With xparse it's easy to do it; the first optional argument has empty default value, whereas we use the \IfNoValueTF test for the second one, in order to avoid an empty superscript which would add an unwanted \scriptspace.

\documentclass{article}
\usepackage{xparse}

\NewDocumentCommand{\der}{O{}mo}{%
  \IfNoValueTF{#3}
    {\frac{\mathrm{d}#1}{\mathrm{d}#2}}
    {\frac{\mathrm{d}^{#3}#1}{\mathrm{d}#2^{#3}}}%
}

\begin{document}
\[
\der[y]{x}\quad
\der[y]{x}[2]\quad
\der[y]{x}[n]\quad
\der{x}\quad
\der{x}[2]\quad
\der{x}[n]
\]
\end{document}