the inverse of a function $H$ is $G$ and assuming that they are both differentiable, then how would you prove that $H'(G(x))G'(x)=1$ for all $x$ in the domain of $G$?
since $x$ is in the domain of $G$ then can i use the inverse function theorem to state that $G'(x)=\frac{1}{H'(G(x))}$?
I wouldn't use it: the theorem about the derivative of the inverse is normally proven using precisely the statement you want to prove here.
Instead, start from $G (F (x))=x $, differentiate both sides, and your statement directly follows (by the chain rule, as Lord Shark has hinted).