Heat capacity and energy dissipation

If we take the definition of the heat capacity as

$$C_v =~~\partial_T ~E~~.$$ (43)

Since the definition of temperature is $\partial_S^{V,n}~ E~=:~T$, which one could arrive at upon fixing V and n in equation (39). Upon using (43), the heat capacity definition could be written as $C_v=\partial_SE~.~\partial_TS~=~T~.~\partial_TS$ . With a little bit of rewriting as

$$C_v~=~T~.~\partial_TS~=~T~(\partial_ES~.~\partial_TE)~=~T~(\frac{1}{T}~.~\partial_TE)$$

and

$$C_v~=~\partial_T^{V,n} E$$ (44)

with the right-hand side as the dissipation term which we have already seen earlier.