I have found some answers here and here, but as they were not explicitly spelt out in the general case I wanted to confirm the following.
I know the derivative of $f(x)$, $\frac{\partial f}{\partial x}=S$ where $f:\mathbb{R}^N\rightarrow \mathbb{R}^N$, exists. I am interested in $\frac{\partial x}{\partial f}$, i.e. the derivative of the inverse function. Unfortunately I cannot simply solve for the inverse function (I might be able to do that, but the resulting format would be difficult to interpret).
Am I correct to say that it would be $\frac{\partial x}{\partial f}=\left(S^T\right)^{-1}$, provided this inverse exists? If it does not exist, could I take the Penrose-Moore Pseudo invserse?
Thanks a lot
Andy