The entropy S(U, V) is a function of U and V in equilibrium. Using the chain rule, dS(U, V) = (dS(U, V)/dU)V dU + (dS(U, V)/dV)U dV
This is always true; it's a mathematical identity.
The "Fundamental Relation of Thermodynamics" says that in equilibrium:
TdS = dU + pdV
where T is temperature, S is entropy, U is internal energy, p is pressure, and V is volume. This describes how, in equilibrium, S depends on changes in U and V.
Here are other derivatives to evaluate, always assuming equilibrium so you can use the Fundamental Relation
1) What's (dS(U, V)/dV)U?
2) What's (dS(U, V)/dU)V?
3) What's (dS(S, V)/dV)S?