Expressions for constant volume and constant pressure heat capacities
The difference in internal energy ($\mathrm{d}U$) is a function of many different variables:

$$\mathrm{d}U = T\,\mathrm{d}S - p\,\mathrm{d}V \!$$

Therefore you must specify – via a partial derivative – with respect to which of the variables you are doing the derivative, otherwise you would need a total derivative, which is not what the equation requires.