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Given H = U + PV and dU = TdS - PdV

Find dH in terms of T, S, P, V

My Solution;

H = U + PV

dH = dU + PdV + VdP

dH = (TdS - PdV) + PdV + VdP

dH = Tds + VdP

My Question

Am I missing a step between the first and second steps? I'm taking the derivative of both sides, but not specifying what the derivative is in respect to. (bad English, sorry)

I learn this short hand from some physics guys, but I'm looking for the strict mathematical method.

Any suggestions?

Thanks,

Bernie