Differentiability of a composite map

Question

Suppose $U,V$ are open subsets of $R^2$ and $S\subseteq R^3$ . Suppose $f:U\to S$ and $g:V\to S$ are bijective differentiable maps whose Jacobians have rank 2. Is the composite map $g^{-1}f$ from $U$ to $V$ a diffeomorphism?