I want to know how to derive pressure $p$ from a partial differential of Helmholtz free energy $F$. I want to start from
$F(T, V, N) = U(S(T,V,N),V, N) - TS(T, V, N)$,
where $U$ and $V$ are internal energy and volume, respectively. Then partial differentiate it and get,
$\left(\frac{\partial F}{\partial V}\right)_{T,N} = \left(\frac{\partial U}{\partial V}\right)_{T,N} - T\left(\frac{\partial S}{\partial V}\right)_{T,N}$.
The LHS should be the $p(T,V,N)$. But I stuck here. Because $\left(\frac{\partial U}{\partial V}\right)_{T,N}$ and $\left(\frac{\partial S}{\partial V}\right)_{T,N}$ is not allowed to express any thermodynamic quantities. (Of course if we fixed $S$ instead of $T$, $\left(\frac{\partial U}{\partial V}\right)_{S,N} = -p(S, V, N)$.) How to go forward?
Thanks in advance!