I'm asked to establish the following relations:
$\left( \frac{\partial C_V}{\partial V} \right)_{T,N} = \frac{T}{N} \left( \frac{\partial^2 P}{\partial T^2} \right)_{V,N} $
$\left( \frac{\partial (\beta F)}{\partial \beta} \right)_{V,N} = U $ where $\beta = \frac{1}{k_B T}$
I'm a bit puzzled by the first one since I can't seem to find a relation between the heat capacity $C_V$ and the other term in the equation, though I do know that $C_V = (\frac{\partial E}{\partial T})_V$
Note that F and U are the Helmholtz free energy and internal energy, respectively.
Also how does the fact that certain quantity are maintained constant throughout the differentiation should be accounted for when expanding say the left hand sides?
Any help would be appreciated.