Second order derivative of the inverse matrix operator 
Let $f : Gl_{n}(\mathbb{R}) \to Gl_{n}(\mathbb{R})$ defined by $f(X)=X^{-1}$. Compute $f''(X)(H,K)$.


I calculated $f'(X).H=-X^{-1}HX^{-1}$ so I tried to use some composition of linear functions but did not find the appropriate functions. Can anyone help me in this matter?

Using the Taylor formula, (As Rodrigo de Azevedo and user1952009 did) you can calculate $f''(X)(H,H)$. If you want the gneral formula $f''(X)(H,K)$, then you can calculate the derivative (with respect to $X$, considering $H$ as a fixed vector) of $f'(X)(H)=-X^{-1}HX^{-1}$. 
We obtain $f''(X)(H,K)=X^{-1}KX^{-1}HX^{-1}+X^{-1}HX^{-1}KX^{-1}$ (that is the "bilinearization" of the quadratic form $f''(X)(H,H)$). 
