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I'm just confused by this, what is the convention for taking these derivatives? For example, if we have S as a function of T, V and P, or S=S(T,V,P)

we want to find partial of S with respect to T, holding P and V constant. Put P and V are functions of T, i.e. P*V=constant*T. How does this work?

If I have, say, S=log(T*V*P), then is the partial derivative mentioned above just equal to (ds/dt)V,P = 1/T ?

What if I write it as S=log(T*(constant*T)) using the relation P*V=constant*T. Shouldn't the derivative be the same thing? Why is there a contradiction?

Does anyone have any good way of explaining this or some good links? I tried to look through books/wikipedia/websites and found nothing.

Thanks.