The function is given by $f(X) = (AX^{-1}A^\top + B)^{-1}$ where $X$, $A$, and $B$ are $n \times n$ positive definite matrices.

I'm trying to find the Lipschitz constant such that $\| f(X)-f(Y) \| \leq L \|X-Y\|$ where $X \geq 0$ and $Y \geq 0$. Motivated by Lemma 3.1 in Nonlinear Systems(H. Khalil, 3rd Ed.), I tried to find the derivative of $f(X)$ (i.e. $\| \frac{ \partial f(X)}{\partial X} \|$) but it's not easy to find the derivative of a function of a matrix over a matrix.

How can I find the Lipschitz constant? or Please let me know if there exists the way to calculate the derivative of a function of a matrix.