Calculating the derivative $\frac{{\partial {{\bf{X}}^{{\rm{ - }}1}}}}{{\partial {\bf{X}}}}$ How to calculate the derivation $\frac{{\partial {{\bf{X}}^{{\rm{ - }}1}}}}{{\partial {\bf{X}}}}$,where ${\bf{X}}$ is square matrix.Thanks a lot for your help! 

If $F(X) = X^{-1}$, the differential of $f$ in $X$ is the linear function given by
$$DF(X)H = - X^{-1}HX^{-1}$$
Proof:
$$F(X+H) - F(X) - DF(X)H = (X+H)^{-1} - X^{-1} + X^{-1}HX^{-1}$$
$$= -(X+H)^{-1}HX^{-1} + X^{-1}HX^{-1} = (X^{-1}-(X+H)^{-1})HX^{-1},$$
so
$$\|F(X+H) - F(X) - DF(X)H\|\le\|X^{-1}-(X+H)^{-1}\|\|H\|\|X^{-1}\| = \|X^{-1}-(X+H)^{-1}\|O(\|H\|).$$
And
$$\lim_{H\to 0}\frac{\|F(X+H) - F(X) - DF(X)H\|}{\|H\|} = 0$$
by the continuity of $F$.
