If, as you say, z= f(g(x,y)) and u= g(x,y), then you have z= f(u) and there are no "partial" derivatives. You should be writing .

since "u" and "g" are equal- they are just different ways of writing the same thing.

In fact, you should not be writing " " because f is a function of thesinglevariable g.

The correct chain rules are and .