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\begin{align*}
        \pder{}{x}\left(-\frac{1}{\rho}\pder{p}{y}\right)&=-\left(\left(\pder{}{x}\frac{1}{\rho}\right)\pder{p}{y}+\frac{1}{\rho}\left(\pder{}{x}\pder{p}{y}\right)\right)\\
        &=-\left(-\frac{1}{\rho^2}\pder{\rho}{x}\pder{p}{y}+\frac{1}{\rho}\pder{^2p}{x\partial y}\right)\\
        \pder{}{y}\left(-\frac{1}{\rho}\pder{p}{x}\right)&=-\left(\left(\pder{}{y}\frac{1}{\rho}\right)\pder{p}{x}+\frac{1}{\rho}\left(\pder{}{y}\pder{p}{x}\right)\right)\\
        &=-\left(-\frac{1}{\rho^2}\pder{\rho}{y}\pder{p}{x}+\frac{1}{\rho}\pder{^2p}{y\partial x}\right)\\
        \pder{}{x}\left(-\frac{1}{\rho}\pder{p}{y}\right)-\pder{}{y}\left(-\frac{1}{\rho}\pder{p}{x}\right)&=-\left(-\frac{1}{\rho^2}\pder{\rho}{x}\pder{p}{y}+\frac{1}{\rho}\pder{^2p}{x\partial y}\right)+\left(-\frac{1}{\rho^2}\pder{\rho}{y}\pder{p}{x}+\frac{1}{\rho}\pder{^2p}{y\partial x}\right)\\
        &=\frac{1}{\rho^2}\left(\pder{\rho}{x}\pder{p}{y}-\pder{\rho}{y}\pder{p}{x}\right)
\end{align*}