To obtain the Maxwell relations we will use the Euler reciprocity relation. If $dz=Mdx+Ndy$ the following relationship holds: \begin{equation} \left(\frac{\partial M}{\partial y}\right)_x=\left(\frac{\partial N}{ \partial x}\right)_y \end{equation} Writing the Gibbs equation for dU: \begin{equation} dU=TdS-PdV=Mdx+Ndy \end{equation} Applying the Euler reciprocity relation we obtain one of Maxwell's equations.
\begin{equation} \left(\frac{\partial T}{\partial V}\right)_s=-\left(\frac{\partial P}{\partial S}\right)_v \end{equation} Applying Euler's relation to the rest of the Gibbs equations, we get: \begin{equation} \left(\frac{\partial T}{\partial P}\right)_s=\left(\frac{\partial V} {\partial S}\right)_p,\;\; \left(\frac{\partial S}{\partial V}\right)_T=\left(\frac{\partial P}{\partial T}\right)_v,\;\; \left(\frac{\partial S}{\partial P}\right)_T=-\left(\frac{\partial V}{\partial T}\right)_p \end{equation}