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For every invertible matrix $A$,The maps $l_C$ and $r_C$ are linear. Differentiate this identity at $A=I$ and use the chain rule and part (a):Thus $Df(C)(Ch)C=-h$. Given an arbitrary increment $k$, set $h=C^{-1}k$ to obtain the derivative of matrix inversionThis also proves differentiability at every $C\in V$.

For a differentiable family of invertible linear operators, differentiate the identity without changing the factor order:Multiplication on the right by $A^{-1}$ gives the derivative of matrix inversion,Commuting $A^{-1}$ past the derivative would generally be invalid.