% !TEX program = pdfLaTeX
\documentclass{article}
\usepackage[margin=3cm]{geometry}
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\input{../00_myPreamble.tex}
\input{../00_myNewCommands.tex}
\begin{document}
\begin{equation}
A = \der{p}{\varrho}{T} ,\quad  
B  = \der{p}{T}{\varrho} ,\quad 
C  = \der{s}{\varrho}{T} ,\quad 
D  = \der{s}{T}{\varrho}
\end{equation}

\begin{equation}
\der{p}{\varrho}{s} = \der{p}{\varrho}{T}- \left. \der{p}{T}{\varrho}\der{s}{\varrho}{T} \middle/ \der{s}{T}{\varrho} \right. = A - \frac{B\cdot C}{D}
\end{equation}

\begin{align}
\der{^2p}{\varrho^2}{s} &= \der{A}{\varrho}{s} -\frac{\der{B}{\varrho}{s} C D + B \der{C}{\varrho}{s} D - B C \der{D}{\varrho}{s} }{D^2} 
\\[10pt] &= \der{A}{\varrho}{s} - \der{B}{\varrho}{s}\frac{C}{D} - \der{C}{\varrho}{s}\frac{B}{D} + \der{D}{\varrho}{s}\frac{BC}{D^2}
\end{align}

\begin{align}
\der{A}{\varrho}{s} &= \der{^2p}{\varrho^2}{T} - \left. \der{^2p}{T\partial\varrho}{} \der{s}{\varrho}{T} \middle/ \der{s}{T}{\varrho} \right. \\
\der{B}{\varrho}{s} &= \der{^2p}{T\partial\varrho}{} - \left. \der{^2p}{T^2}{\varrho} \der{s}{\varrho}{T} \middle/ \der{s}{T}{\varrho} \right. \\
\der{C}{\varrho}{s} &= \der{^2s}{\varrho^2}{T} - \left. \der{^2s}{T\partial\varrho}{} \der{s}{\varrho}{T} \middle/ \der{s}{T}{\varrho} \right. \\
\der{D}{\varrho}{s} &= \der{^2s}{T\partial\varrho}{} - \left. \der{^2s}{T^2}{\varrho} \der{s}{\varrho}{T} \middle/ \der{s}{T}{\varrho} \right. 
\end{align}

\begin{equation}
\begin{split}
\der{^2p}{\varrho^2}{s} = \der{^2p}{\varrho^2}{T} 
&- \left. \left[\der{p}{T}{\varrho} \der{^2s}{\varrho^2}{T} + 2 \der{^2p}{T\partial\varrho}{} \der{s}{\varrho}{T} \right] \middle/ \der{s}{T}{\varrho} \right. \\
&+ \left. \left[ \der{^2p}{T^2}{\varrho} \der{s}{\varrho}{T}^2 + 2 \der{p}{T}{\varrho} \der{s}{\varrho}{T} \der{^2s}{T\partial\varrho}{} \right] \middle/ \der{s}{T}{\varrho}^{2} \right. \\
&- \left. \left[ \der{p}{T}{\varrho} \der{s}{\varrho}{T}^2 \der{^2s}{T^2}{\varrho} \right] \middle/ \der{s}{T}{\varrho}^{3} \right.
\end{split}
\end{equation}
\end{document}