Given the equation $U=T\frac{\partial{S}}{\partial{T}}dT$ where $S$ is a function of V and T and V is being held constant, how do we then find $\frac{\partial U}{\partial T}$?
The way I'm explaining this to myself is that it's essentially the same as in single variable calculus when we have some term multiplied by a differential term (which in this case is the factor $dT$), and we want to differentiate with respect to the variable of the differential term ($T$ here), all that happens is that the differential term disappears.
So the answer would be $\frac{\partial U}{\partial T}=T\frac{\partial{S}}{\partial{T}}$
Is this correct? What is happening here? Is the chain rule being used subtly? Are there any complications here due to the partial derivatives as opposed to the single variable version?
I realise this isn't a clearly defined question, I'm just hoping for some helpful and relevant remarks.