How to write partial differential equation (Ex. dQ/dt=ds/dt) with real partial derivative signs?

You said partial differential equation:

\documentclass{article}
\begin{document}
\begin{equation}
\frac{\partial Q}{\partial t} = \frac{\partial s}{\partial t}
\end{equation}
\end{document}

now using physics package, extra goodies (bonus):

\documentclass{article}
\usepackage{physics}
\begin{document}
\[
\dv{Q}{t} = \dv{s}{t}  \quad
\dv[n]{Q}{t} = \dv[n]{s}{t}  \quad
\pdv{Q}{t} = \pdv{s}{t}  \quad
\pdv[n]{Q}{t} = \pdv[n]{s}{t}  \quad
\pdv{Q}{x}{t} = \pdv{s}{x}{t}  \quad
\]
\[
\fdv{F}{g}
\]
\end{document}

I now recommend using the cool package:

\documentclass{article}
\usepackage{cool}
\begin{document}
Text:
\[
    \pderiv{u}{t}=\pderiv[2]{u}{x}
\]
More text.
\end{document}

I used to recommend defining a command to make a short form:

\documentclass{article}
% Partial derivative
\newcommand*{\pd}[3][]{\ensuremath{\frac{\partial^{#1} #2}{\partial #3}}}
\begin{document}
Text:
\[
    \pd{u}{t}=\pd[2]{u}{x^2}
\]
More text.
\end{document}

Another possibility to write classic derivates or partial derivates I suggest (IMHO), actually, to use derivative package. For my humble opinion it is very good and last release is v0.95b 2019/09/21. Here there are some examples take, some, from the guide:

\documentclass[12pt]{article}
\usepackage{derivative}
\begin{document}
\[
\pdv{f}{x}, \quad \odv{Q}{t}=\odv{s}{t}, \quad \pdv{f}{x,y}, \quad \derivset{\odv}[switch-*=false] \odv{y}{x}, \quad \odv[n]{y}{x}, \quad \derivset{\odv}[misc-add-delims=fun] \odv*{\odv{y}{x}}{x}, \quad \derivset{\pdv}[sort-method={sign,symbol,abs}] \pdv[c+kn,-b+2a]{f}{x,y}
\]
\end{document}