Maxwell relations are derived from the fact that thermodynamic potentials (like $U$) are state functions with exact differentials.

• From the first law: $dU=TdS-PdV$.
• Since $dU$ is an exact differential, the mixed partial derivatives must be equal: ${\left(\frac{\partial T}{\partial V}\right)}_S={\left(\frac{\partial (-P)}{\partial S}\right)}_V=-{\left(\frac{\partial P}{\partial S}\right)}_V$ This identifies a deep connection between the temperature-volume relationship and the pressure-entropy relationship.