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I have a question regarding differentiation of inverse functions that I am not capable of solving. I want to prove that

[itex]\frac{\partial}{\partial y} h_y(h^{-1}_{y_0}(z_0))\bigg|_{y=y_0} = - \frac{\partial}{\partial y} h_{y_0}(h^{-1}_{y}(z_0))\bigg|_{y=y_0},[/itex]

where

[itex]h_y(x)[/itex] is considered as a function of x with a secondary variable y attached.

[itex]h^{-1}_y(z)[/itex] is the inverse function of [itex]h[/itex] written as a function of z, of course also depending of [itex]y[/itex], precisely given as the to solution to [itex]z=h_y(x)[/itex].

I have tested the relation with a wide range of easy-to-check h-functions and it holds in all cases I have checked. By using the chain rule I could rewrite the right-hand side to a quantity easier to handle, but as I am not able to re-write the left-hand side in any way this does not really help me.

As the derivative is with respect to the secondary y-variable and not the variable that that the inverse is taken with respect to I cannot apply the rule for differentiating inverse functions either.

Does anyone have any clue how I might prove that this holds, or have a counter-example showing that it does not hold? Any help or pointing to references are highly appreciated.

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# Differentiation of a two dimensional inverse function

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