Is there a short hand command to write derivatives?

You can use the esdiff package, which has handy macros for derivatives and partial derivatives, taking care of indices. Here is a demo;

\documentclass{article}%

\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{mathtools}
\usepackage[thinc]{esdiff}

 \begin{document}

\begin{alignat*}{3}
     \diff{f}{x} &\qquad \diff*[4]{g}{t}{t = 1} \\[2ex]
     \diffp{f}{x} &\qquad\diffp{g}{tu}& & \qquad & \diffp*{g}{{t^2}{u^3}}{(0,0)}
 \end{alignat*}

 \end{document}

Oh, you mean not symbol, but operator. There is physics, as stated by @Zarko.

• \differential produces the variants of d: \dd x
• \derivative yields the $df/dx$ in variants: \dv{x}, \dv{f}{x}
• \partialderivative produces the partial symbol in derivaties a la carte, similar to \dv, use \pdv x, etc.
• \variation and \functionalderivative are also there, e.g. \fdv{F}{g}.

It's all in the documentation on pages 5-6, say texdoc physics for the pdf.

---

Screenshot from the document.

There is also another important and easy package to write ordinary derivate and partial derivatives named derivative. I have added only some simple examples how to use this package where the d ("classical derivate") is written in roman.

\documentclass[12pt]{article}
\usepackage{derivative}
\usepackage{amsmath,amssymb}
\begin{document}

\[\odv{f(x)}{x}, \qquad \odv[n]{f(x)}{x}, \qquad \pdv[\alpha,b,c]{f}{x,y,z}\]
\[\pdv{f}{x,y}, \qquad \odv{e^{\tan(x)}}{x}, \qquad \odv{\cos t}{t}_{t=0}^{}\]
\end{document}