# Differential and derivative of $X^{-2}$

Determine the differential and derivative of $F(X) = X^{-2}$ in which the variable X is an n x n-matrix.

I computed the differential by using the product rule. So I first wrote $$f(X)= X^{-1} X^{-1}$$, so $$d(f(x))= d(X^{-1})(X^{-1}) + (X^{-1})d(X^{-1}) = (-X^{-1})d(X)(X^{-1})(X^{-1}) -(X^{-1})(X^{-1})d(X)(X^{-1}).$$ Did I compute this correctly? And how do I find the derivative now? Thanks a lot in advance.

Edit: I computed the derivative and I think it should be -((X^-2)' tensor (X^-1)) -((X^-1)'tensor(X^-2)). Could anyone give any feedback on whether I did this correctly (a prime denotes a transpose)?

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Your map $f$ is the composition of $g(X)=X^2$ and $h(X)=X^{-1}$.

The differential of the first one is $$dg_X(H)=XH+HX$$ and for the second one, it is $$dh_X(H)=-X^{-1}HX^{-1}.$$

By the chain rule, $$df_X(H)=dh_{g(X)}\circ dg_X(H)=dh_{X^2}(XH+HX)$$ $$=-X^{-2}(XH+HX)X^{-2}=-X^{-1}HX^{-2}-X^{-2}HX^{-1}.$$

So yes, you are correct. But your notations are slightly confusing. Only slightly.

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Hahaha ok, thanks a lot. And now how do I find the derivative? By vectorising? – dreamer Feb 24 '13 at 15:28
Could you please check if I computed the derivative correctly? – dreamer Feb 24 '13 at 15:35
@user48288 Yes, you are correct. Provided $-X^{-1}d(X)X^{-2}$ means $H\longmapsto -X^{-1}HX^{-2}$. But I believe that's what you meant. – 1015 Feb 24 '13 at 15:37
Yes thats what I meant. Thanks a lot. I think I finally start to become acquainted with computing these kinds of differentials and derivatives :). Thanks a lot again for all your help recently :). – dreamer Feb 24 '13 at 15:40
@user48288 You're most welcome. – 1015 Feb 24 '13 at 15:41