\[\begin{align} d S &=\frac{1}{T}[d U+P d V] \\[4pt] &=\dfrac{1}{T}\left[\left(\frac{\partial U}{\partial V}\right)_{T} d V+\left(\frac{\partial U}{\partial T}\right)_{V} d T+P d V\right].\\[4pt] &=\f...\[\begin{align} d S &=\frac{1}{T}[d U+P d V] \\[4pt] &=\dfrac{1}{T}\left[\left(\frac{\partial U}{\partial V}\right)_{T} d V+\left(\frac{\partial U}{\partial T}\right)_{V} d T+P d V\right].\\[4pt] &=\frac{1}{T}\left[\left(\frac{\partial U}{\partial V}\right)_{T}+P\right] d V+\frac{1}{T}\left(\frac{\partial U}{\partial T}\right)_{V} d T. \end{align}\]
