\begin{equation}
E=\sum_{i}N_{i}\epsilon_{i}=\sum_{i}\frac{N}{q}g_{i}e^{-\beta{\epsilon_{i}}}\epsilon_{i}=\frac{N}{q}\sum_{i}g_{i}\epsilon_{i}e^{-{\epsilon_{i}}/kT}
\label{ec31}
\end{equation}
Derivando $q=\sum_{i}g_{i}e^{-\epsilon_{i}/kT}$ respecto de $T$ a volumen:
\begin{equation}
\left(\frac{\partial{q}}{\partial{T}}\right)_{V}=\frac{1}{kT^2}\sum_{i}g_{i}\epsilon_{i}e^{-\epsilon_{i}/kT}
\label{ec32}
\end{equation}
Despejando el sumatorio de la ecuación (~\ref{ec32}):
\begin{equation}
kT^{2}\left(\frac{\partial{q}}{\partial{T}}\right)_{V}=\sum_{i}g_{i}\epsilon_{i}e^{-\epsilon_{i}/kT}
\label{ec33}
\end{equation}
Sustituyendo (~\ref{ec33}) en (~\ref{ec31})
\begin{equation}
E=NkT^{2}\left(\frac{\partial{lnq}}{\partial{T}}\right)_{V}
\label{ec34}
\end{equation}