Derivative of matrix inversion function? Let's say I have a function $f$ which maps any invertible $n\times n$ matrix to its inverse. How do I calculate the derivative of this function?

$$(X+\delta)^{-1} = [X\cdot ( I + X^{-1} \cdot \delta)]^{-1}= ( I + X^{-1} \cdot \delta)^{-1}\cdot X^{-1} = \\
=(I - X^{-1} \delta + X^{-1} \delta X^{-1} \delta - \ldots ) \cdot X^{-1}= X^{-1} - X^{-1} \delta X^{-1} +  X^{-1} \delta X^{-1} \delta X^{-1} - \ldots 
$$
so the linear part of the variation is $-X^{-1} \delta X^{-1}$.
We also have the Taylor series expansion at $X$.
