% !TeX root = v_rho_rewrite_Standalone.tex
% !TeX program = pdfLaTeX
{\setlength{\tabcolsep}{.2em}
\begin{tabularx}{\linewidth}{ccl}
\toprule
$\der{a}{v}{c}$ & $=$ & $-\varrho^2 \der{a}{\varrho}{c}$ \\[12pt]
$\der{a}{\varrho}{c}$ & $=$ & $-v^2 \der{a}{v}{c}$ \\
\midrule
$\der{v}{b}{c}$ & $=$ & $-\dfrac{1}{\varrho^2} \der{\varrho}{b}{c}$ \\[12pt]
$\der{\varrho}{b}{c}$ & $=$ & $-\dfrac{1}{v^2} \der{v}{b}{c}$ \\
\midrule[1pt]
%$\der{^2 p}{v^2}{T}$ %& $=$ & $ \left( \dfrac{\partial}{\partial v} \der{p}{v}{T} \right)_{T} = -\varrho^2 \left( \dfrac{\partial}{\partial \varrho} \left( -\varrho^2 \left(\dfrac{\partial p}{\partial \varrho} \right)_{T} \right) \right)_{T} $\\[12pt]
% & $=$ &$2\varrho^3 \der{p}{\varrho}{T} + \varrho^4 \der{^2 p}{\varrho^2}{T}$ \\[12pt]
%\midrule
%$\der{^2 p}{v^2}{s}$ & $=$ & $2\varrho^3 \der{p}{\varrho}{s} + \varrho^4 \der{^2 p}{\varrho^2}{s}$ \\[12pt]
%$\der{^2 p}{\varrho^2}{s}$ & $=$ & $2v^3 \der{p}{v}{s} + v^4 \der{^2 p}{v^2}{s}$ \\
%\midrule
$\der{^2 a}{v^2}{c}$ & $=$ & $2\varrho^3 \der{a}{\varrho}{c} + \varrho^4 \der{^2 a}{\varrho^2}{c}$ \\[12pt]
$\der{^2 a}{\varrho^2}{c}$ & $=$ & $2v^3 \der{a}{v}{c} + v^4 \der{^2 a}{v^2}{c}$ \\
\midrule
$\der{^2 v}{b^2}{c}$ & $=$ & $\dfrac{2}{\varrho^3} \der{\varrho}{b}{c} - \dfrac{1}{\varrho^2} \der{^2 \varrho}{b^2}{c}$ \\[12pt]
$\der{^2 \varrho}{b^2}{c}$ & $=$ & $\dfrac{2}{v^3} \der{v}{b}{c} - \dfrac{1}{v^2} \der{^2 v}{b^2}{c}$ \\
\bottomrule
\end{tabularx}%
}