In this section we will obtain the expressions for the differentials dH, dA, dG using the relation $dU=TdS-PdV$ from the definitions $H=U+PV$, $A=U-TS$ and $G=H- TS$.
We start by getting dH: \begin{equation} dH=d(U+PV)=dU+PdV+VdP=TdS-PdV+PdV+VdP=TdS+VdP \end{equation} Similarly we get dA and dG. \begin{equation} dA=d(U-TS)=dU-SdT-TdS=\cancel{TdS}-PdV-SdT-\cancel{TdS}=-SdT-PdV \end{equation} \begin{equation} dG=d(H-TS)=dH-TdS-SdT=\cancel{TdS}+VdP-\cancel{TdS}-SdT=-SdT+VdP \end{equation} Therefore the Gibbs equations are: \begin{eqnarray} dU & = & TdS - PdV\\ dH & = & TdS+VdP\\ dA & = & -SdT-PdV\\ dG & = & -SdT+VdP \end{eqnarray}
From equation (29) the following relations can be obtained: \begin{equation} \left(\frac{\partial U}{\partial S}\right)_v=T\;\;\; \left(\frac{\partial U}{\partial V}\right)_s=-P \end{equation} The first relation is obtained by making dV=0 in (29), and the second with dS=0.
From equation (30) we obtain: \begin{equation} \left(\frac{\partial H}{\partial S}\right)_p = T\;\;\; \left(\frac{\partial H}{\partial P}\right)_s = V \end{equation} From equation (31) we obtain: \begin{equation} \left(\frac{\partial A} {\partial T}\right)_v = -S\;\;\; \left(\frac{\partial A}{\partial V}\right)_T = -P \end{equation} From equation (32) we obtain: \begin{equation} \left(\frac{\partial G }{\partial T}\right)_p = -S\;\;\; \left(\frac{\partial A}{\partial P}\right)_T = V \end{equation} The objective of these equations is to relate thermodynamic properties that are difficult to obtain with others that are easily measurable, such as: $C_p$, $\alpha$ and $\kappa$.