Derivative of Inverse of sum of matrices

Given is the function $f : mathbb{R}^p to mathbb{R}$ with

$$
f(x) = q(x)^{top} G^{-1} q(x)
$$

where $G = A + x_1 B_1 + ldots + x_p B_p$.

• The matrices $A, B_1, ldots, B_p in mathbb{R}^{n times n}$ are all symmetric positive definit.

• $q: mathbb{R}^n to mathbb{R}^n$ and the Jacobian $nabla q$ is known.

Is it possible to derive a closed form for $nabla f$? For me, the hard part is
$G^{-1}$. Any hints or suggestions are really appreciated!