I was asked to use direct differentiation of "H = U + pV" to find a relationship between:
$$ \left(\frac{\partial H}{\partial U}\right)_{\!P}$$ and $$\left(\frac{\partial U}{\partial V}\right)_{\!P} $$
So my first step was to take a differential of the equation twice, once in terms of U and once in terms of V:
$$ \left(\frac{\partial}{\partial U}H\right)_{\!P}=\left(\frac{\partial}{\partial U}(U +PV)\right)_{\!P}\rightarrow\left(\frac{\partial H}{\partial U}\right)_{\!P}= 1 + P\left(\frac{\partial V}{\partial U}\right)_{\!P}$$ and
$$ \left(\frac{\partial}{\partial V}H\right)_{\!P}=\left(\frac{\partial}{\partial V}(U +PV)\right)_{\!P}\rightarrow\left(\frac{\partial H}{\partial V}\right)_{\!P}= \left(\frac{\partial U}{\partial V}\right)_{\!P} + P$$
When I tried to substitute one equation into the other, they would simply end up all canceling out. Is there another way?