= Solution
Differentiate $H=U+pV$ at fixed <pressure> and subtract the <heat capacity at constant volume>:
$$
C_p-C_V=\left(\frac{\partial U}{\partial T}\right)_p+p\left(\frac{\partial V}{\partial T}\right)_p-\left(\frac{\partial U}{\partial T}\right)_V.
$$
Now use $U=U(T,V)$ and the <multivariable chain rule> along a constant-pressure path:
$$
\left(\frac{\partial U}{\partial T}\right)_p=\left(\frac{\partial U}{\partial T}\right)_V+\left(\frac{\partial U}{\partial V}\right)_T\left(\frac{\partial V}{\partial T}\right)_p.
$$
Substitution cancels the fixed-volume <derivative> and gives the <volume derivative form of the heat-capacity difference>,
$$
\boxed{C_p-C_V=\left[\left(\frac{\partial U}{\partial V}\right)_T+p\right]\left(\frac{\partial V}{\partial T}\right)_p.}
$$
For the <ideal gas>, the first term in brackets vanishes and $V=NkT/p$, so $(\partial V/\partial T)_p=Nk/p$. Consequently $\boxed{C_p-C_V=Nk}$ for the entire gas, with fixed particle number $N$. The extra <heat capacity> accounts for expansion <work> at constant <pressure>.
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