= Solution
The required <Legendre transform> is the <Helmholtz free energy>, $\boxed{F=U-TS}$ up to an additive constant. Its differential is $dF=dU-TdS-SdT=-SdT-pdV$, so $F_T|_V=-S$ and $F_V|_T=-p$. Equality of mixed partial derivatives for a smooth thermodynamic state gives the <Maxwell relation>
$$
\boxed{\left(\frac{\partial S}{\partial V}\right)_T=\left(\frac{\partial p}{\partial T}\right)_V.}
$$
Next use the <total differential> $dS=S_T|_VdT+S_V|_TdV$. Along a fixed-pressure path, divide by $dT$ to get $S_T|_p=S_T|_V+S_V|_T V_T|_p$. Substitute the Maxwell relation to obtain
$$
\boxed{\left(\frac{\partial S}{\partial T}\right)_p-\left(\frac{\partial S}{\partial T}\right)_V=\left(\frac{\partial p}{\partial T}\right)_V\left(\frac{\partial V}{\partial T}\right)_p.}
$$
Multiplying by $T$ gives the usual difference between the <heat capacity at constant pressure> and the <heat capacity at constant volume>.
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