Definition of specific heat by via entropy

[mSSM]

In his Statistical Physics book, Landau introduces the specific heat as the quantity of heat which must be gained in order to raise the temperature of a body one by unit.

I don't understand, how he directly jumps to the conclusion that that has to be (let's just say, for constant volume):
$$C_V = T\left(\frac{\partial S}{\partial T}\right)_V$$

If I take a process during which I have no change in volume, I think I can write: $\partial Q = T\mathrm{d}S$, is that correct? My next thought was then that I could write $C_V=\left(\frac{\partial Q}{\partial T}\right)_V$. However, now I am stuck, because I do not understand why this should yield the above definition, without acting on the $T$ part of the the heat-entropy relation.EDIT: I would like to add, that I have the same problem with the specific heat's definition via the energy differential $\mathrm{d}E = T\mathrm{d}S - P\mathrm{d}V$:
$$C_V = \left(\frac{\partial E}{\partial T}\right)_V$$

Why does $\frac{\partial}{\partial T}$ not act on $T$?

 

[Chestermiller]

You need to go back and learn how partial differentiation is done. This is a mathematics question, not a physics question.