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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

Andy
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    Do You know the inverse function theorem, look e.g. https://en.wikipedia.org/wiki/Inverse_function_theorem – Peter Melech Feb 20 '18 at 09:39
  • The Jacobian of the inverse is the inverse of the Jacobian. – Arthur Feb 20 '18 at 09:42
  • Thank you. I did look at the inverse function theorem and was a bit confused as in one of the answers here it looked like you have to transpose the Jacobian. If the inverse of the Jacobian does not exist (unlikely in my case, but just to be sure), would using the pseudo-inverse work? – Andy Feb 20 '18 at 13:19
  • @Andy why would it? The inverse function might be not differentiable. – user251257 Feb 20 '18 at 19:48
  • @user251257: Silly me, of course. I checked, it is differentiable in all relevant cases (yes, there are instances where it is not invertible, but they are irrelevant luckily) – Andy Feb 21 '18 at 11:40

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