Differentiation of H = U + pV
A way to do it would be:

\begin{align}
\left( \frac{\partial H}{\partial U} \right)_P \left( \frac{\partial U}{\partial V} \right)_P &amp;= \left( \frac{\partial H}{\partial V} \right)_P = \left( \frac{\partial U}{\partial V} \right)_P + P\\
\left( \frac{\partial H}{\partial U} \right)_P &amp;= \frac{\left( \frac{\partial U}{\partial V} \right)_P + P}{\left(\frac{\partial U}{\partial V} \right)_P}\\
\left(\frac{\partial U}{\partial V} \right)_P &amp;= \frac{P}{\left( \frac{\partial H}{\partial U} \right)_P - 1}\\
\end{align}