Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 2.3-3

The fundamental equation gives you $u$ as a function of $s$ and $v$, which means you're working in the entropy representation where internal energy is the characteristic function. To find the three equations of state, use the definitions of temperature, pressure, and chemical potential as partial derivatives of $u$: $T = (\partial u/\partial s)_v$, $P = -(\partial u/\partial v)_s$, and $\mu = u$ itself (or derive it from the Gibbs-Duhem relation). Each partial derivative will give you a relationship between state variables that describes the system's behavior.

The fundamental equation gives you $u$ as a function of $s$ and $v$, which means you're working in the entropy representation where internal energy is the characteristic function. To find the three equations of state, use the definitions of temperature, pressure, and chemical potential as partial derivatives of $u$: $T = (\partial u/\partial s)_v$, $P = -(\partial u/\partial v)_s$, and $\mu = u$ itself (or derive it from the Gibbs-Duhem relation). Each partial derivative will give you a relationship between state variables that describes the system's behavior.