Imagine I have a vector field $f^a(x,y)$ that depends on vector positions $x^i$ and $y^i$ with $i=1,\dots,N$. The index $a=1,\dots,M$. Now, let's say There's another function $G: R^M\rightarrow R$ and I want to evaluate
\begin{equation} \frac{\partial}{\partial x^i} G(f^a(x,y)) \end{equation}
This can be easily done with the chain rule. We get
\begin{equation} \frac{\partial}{\partial x^i} G(f^a(x,y))= \frac{\partial f^a(x,y)}{\partial x^i}\frac{\partial}{\partial u^a}G(u) \end{equation}
My question is: Is there a way to to the opposite calculation to get
\begin{equation} \frac{\partial}{\partial u^a}G(u) =\big[\frac{\partial f^a(x,y)}{\partial x^i}\big]^{-1}\frac{\partial}{\partial x^i} G(f^a(x,y)) \end{equation}
and when is this well defined?