Expressions for Heat Capacity

Homework Statement

Let $C_{X}=T(\partial S/\partial T)_{X}$ with $X=p~or~V$. Show that $C_{p}-C_{V}=T(\partial S/\partial V)_{T}(\partial V/\partial T)_{p}$.

Homework Equations

$C_{p}-C_{V}=T[(\partial S/\partial T)_{p}-(\partial S/\partial T)_{V}]$

The Attempt at a Solution

I honestly don't know where to start. I looked at some of the Maxwell relations, but no Maxwell relation features conjugate variables in the numerator and denominator of a partial derivative. I'm not sure how what other way I might use to manipulate the thermodynamic variables.