= Solution
For a fixed number of non-interacting molecules, $Z_N=q^N/N!$ in the dilute <ideal gas> limit. The temperature-independent factorial drops out of the <internal energy> derivative:
$$
U_m=RT^2\left(\frac{\partial\ln q}{\partial T}\right)_V.
$$
Differentiate once more at fixed volume to obtain
$$
\boxed{\frac{C_{V,m}}R=2T\frac{q'}q
+T^2\left[\frac{q''}q-\left(\frac{q'}q\right)^2\right]
=\frac{d}{dT}\left(T^2\frac{d\ln q}{dT}\right).}
$$
Primes denote fixed-volume derivatives. If $q$ factorizes into translational, rotational and other independent contributions, their <heat capacities> add. In particular, the three-dimensional translational factor contributes $3R/2$, while a constant <spin degeneracy> factor contributes zero. The formula also follows from the positive <canonical energy variance>, $C_V=\operatorname{Var}(E)/(k_{\rm B}T^2)$.
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