[UPSC IFoS 2017] Thermodynamics (Q70, 8M)

Prove the thermodynamic relation : \left( \frac{\partial S}{\partial V} \right)_T = \left( \frac{\partial P}{\partial T} \right)_V and hence show that \frac{\mathrm{d}P}{\mathrm{d}T} = \frac{L}{T (V_2 - V_1)} ; all the terms have their usual meanings.