Adding parameters, inside a newcommand for partial derivatives

macrosmath-modexfp

Using \usepackage{xfp}, I have created the following command:

\newcommand{\ppdv}[5]{\dfrac{\partial^{\fpeval{#3 + #5}} #1}{\partial {#2}^{#3} \partial {#4}^{#5}}}

So that, for example, if I type: \ppdv{f}{x}{2}{y}{3}
I will get: \dfrac{\partial^5 f}{\partial x^2 \partial y^3}

I have two problems:

If I write \ppdv{f}{x}{2}{y}{}, meaning I only want to take the first derivative of variable y (I don't want to write number 1),

I won't get: \dfrac{\partial^3 f}{\partial x^2 \partial y}

And if I write \ppdv{f}{x}{n}{y}{4}, meaning I want to calculate the n-th derivative of variable x,

I won't get: \dfrac{\partial^{n+4} f}{\partial x^2 \partial y}

Tex doesn't compile these tasks using xfp package. Is there a way to solve this problem? I won't bother changing xfp package for another one anyway.

Also, If I write, for example, \ppdv{f}{x}{2}{y}{m}, meaning I want to calculate the m-th derivative of variable y,

I would like to get, if possible: \dfrac{\partial^{m+2} f}{\partial x^2 \partial y} with de unknown letter m, being the first one to appear in the sum.

If anyone knows how to solve this problem I will appreciate it.

Best Answer

Here is one way to do it:

enter image description here

Code:

\documentclass{article}
\usepackage{mathtools}
\usepackage{xfp}
%\usepackage{pgfmath}
\usepackage{xstring}

\makeatletter
\def\@IntergerSum{0}%
\def\@NonIntergerSum{}%
\newcommand{\@GetSumAux}[1]{%
    \IfStrEq{#1}{}{%
        %% Ignore if this is empty
    }{%
        \IfInteger{#1}{%
            %\pgfmathtruncatemacro\@IntergerSum{\@IntergerSum + #1}% pgfmath version
            \edef\@IntergerSum{\fpeval{\@IntergerSum + #1}}%         xfp version
        }{%
            \IfStrEq{\@NonIntergerSum}{}{%
                \g@addto@macro\@NonIntergerSum{#1}%
            }{%
                \g@addto@macro\@NonIntergerSum{+ #1}%
            }%
        }%
    }%
}%
\newcommand{\@GetSum}[5]{%
    %% If inputs are integers their values are summed, otherwise we return
    %% an expression for the sum. Thus, if the last four parameters are
    %%
    %%      {1}{2}{3}{4} = 10
    %%      {1}{2}{3}{n} = n + 6
    %%      {1}{m}{3}{n} = m + n + 4
    %%
    %% #1 = macro to set
    %% #2 = parameter
    %% #3 = parameter
    %% #4 = parameter
    %% #5 = parameter
    %% -------------
    \gdef\@IntergerSum{0}%
    \gdef\@NonIntergerSum{}%
    %% ----------
    \@GetSumAux{#2}%
    \@GetSumAux{#3}%
    \@GetSumAux{#4}%
    \@GetSumAux{#5}%
    \IfStrEq{\@NonIntergerSum}{}{%
        %\edef#1{\@IntergerSum}%
        \@GetExponent{#1}{\@IntergerSum}%
    }{%
        \IfStrEq{\@IntergerSum}{0}{%
            \edef#1{^{\@NonIntergerSum}}%
        }{%
            \edef#1{^{\@NonIntergerSum + \@IntergerSum}}%
        }%
    }%
}%
\newcommand{\@GetExponent}[2]{%
    %% hides exponent of 1
    %% #1 = macro to set
    %% #2 = value
    \IfStrEq{#2}{}{%
        \def#1{}%%% No exponent if this is empty
    }{%
        \IfStrEq{#2}{1}{%
            \def#1{}% Suppress exponent of 1
        }{%
            \def#1{^{#2}}%
        }%
    }%
}%
\newcommand{\@ShowVar}[2]{%
    %% #1 = variable
    %5 #2 = exponenet
    \IfStrEq{#1}{}{}{\partial#1#2}% %% If no var given, we skip this variable
}%
\newcommand*{\@SumExponenet}{}%
\newcommand{\@PPDV}[9]{%
    \begingroup
    \@GetSum{\@SumExponenet}{#3}{#5}{#7}{#9}%
    %% ----------
    \@GetExponent{\@ExponenentV}{#3}%
    \@GetExponent{\@ExponenentX}{#5}%
    \@GetExponent{\@ExponenentY}{#7}%
    \@GetExponent{\@ExponenentZ}{#9}%
    %% ----------
    \dfrac
        {\partial\@SumExponenet#1}
        { 
            \@ShowVar{#2}{\@ExponenentV}%
            \@ShowVar{#4}{\@ExponenentX}%
            \@ShowVar{#6}{\@ExponenentY}%
            \@ShowVar{#8}{\@ExponenentZ}%
        }%
    \endgroup
}

\newcommand{\pdv}[3]{\@PPDV{#1}{#2}{#3}{}{}{}{}{}{}}
\newcommand{\ppdv}[5]{\@PPDV{#1}{#2}{#3}{#4}{#5}{}{}{}{}}
\newcommand{\pppdv}[7]{\@PPDV{#1}{#2}{#3}{#4}{#5}{#6}{#7}{}{}}%
\newcommand{\ppppdv}[9]{\@PPDV{#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}{#9}}%
\makeatother


\begin{document}
\begin{gather*}
    \ppdv{f}{x}{1}{y}{1}% test exponent of 1
    \;
    \ppdv{f}{x}{2}{y}{3}% all numerical not 1
    \;
    \ppdv{f}{x}{n}{y}{4}% first non-numerical
    \;
    \ppdv{f}{x}{2}{y}{m}% second nun-numerical (desired output "m+2", not "2+m")
    \;
    \ppdv{f}{x}{n}{y}{m}% bot non numerical
    \\[2.0ex]
    \pppdv{f}{x}{1}{y}{1}{z}{1}% test exponent of 1
    \;
    \pppdv{f}{x}{2}{y}{3}{z}{4}% all numerical not 1
    \;
    \pppdv{f}{x}{m}{y}{1}{z}{2}% first non-numerical
    \;
    \pppdv{f}{x}{1}{y}{n}{z}{3}% second non-numerical
    \\[0.5ex]
    \pppdv{f}{x}{1}{y}{2}{z}{p}% third non-numerical
    \;
    \pppdv{f}{x}{m}{y}{2}{z}{p}% first & third  non-numerical
    \;
    \pppdv{f}{x}{2}{y}{m}{z}{p}% second & third  non-numerical
    \\[2.0ex]
    \ppppdv{f}{u}{1}{x}{1}{y}{1}{z}{1}% test exponent of 1
    \;
    \ppppdv{f}{u}{1}{x}{n}{y}{m}{z}{1}% second and third exponent of 1
    \;
    \ppppdv{f}{u}{m}{x}{n}{y}{p}{z}{r}% all non-numerical
\end{gather*}
\end{document}
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