2) This problem asks you to derive some derivative identities of a system with three variables \( x, y \) and \( z \), with a single constraint \( x(y, z) \). This kind of system is central to thermodynamics as we often use three state variables \( P, V \) and \( T \), with an equation of state \( P(V, T) \) (i.e. the constraint) to describe a system.
(i) Prove the following identity
\[
\left(\frac{\partial x}{\partial y}\right)_{z}\left(\frac{\partial y}{\partial z}\right)_{x}\left(\frac{\partial z}{\partial x}\right)_{y}=-1
\]
and hence show that for any equation of state \( P(V, T) \) with the state variables \( P, V \) and \( T \), the partial derivatives are related by
\[
\left(\frac{\partial V}{\partial T}\right)_{P}\left(\frac{\partial T}{\partial P}\right)_{V}\left(\frac{\partial P}{\partial V}\right)_{T}=-1
\]
