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?

Let $f (\mathrm X) := \mathrm X^{-1}$. The 1st directional derivative of $f$ in the direction of $\mathrm V_1$ at $\mathrm X$ is $\color{blue}{-\mathrm X^{-1} \mathrm V_1 \mathrm X^{-1}}$.
Let $\mathrm G (\mathrm X) := -\mathrm X^{-1} \mathrm V_1 \mathrm X^{-1}$. Hence,
$$\mathrm G (\mathrm X + h \mathrm V_2) \approx \mathrm G (\mathrm X) + h \left( \mathrm X^{-1} \mathrm V_1 \mathrm X^{-1} \mathrm V_2 \mathrm X^{-1} + \mathrm X^{-1} \mathrm V_2 \mathrm X^{-1} \mathrm V_1 \mathrm X^{-1} \right)$$
Thus, the 1st directional derivative of $\mathrm G$ in the direction of $\mathrm V_2$ at $\mathrm X$ is
$$\color{blue}{\mathrm X^{-1} \mathrm V_1 \mathrm X^{-1} \mathrm V_2 \mathrm X^{-1} + \mathrm X^{-1} \mathrm V_2 \mathrm X^{-1} \mathrm V_1 \mathrm X^{-1}}$$
which is the 2nd directional derivative of $f$ in the directions of $\mathrm V_1$ and $\mathrm V_2$ at $\mathrm X$. If $\mathrm V_1 = \mathrm V_2 = \mathrm V$, then the 2nd directional derivative of $f$ in the direction of $\mathrm V$ at $\mathrm X$ boils down to $\color{blue}{2 \mathrm X^{-1} \mathrm V \mathrm X^{-1} \mathrm V \mathrm X^{-1}}$.
