Given a differentiable $A: \mathbb{R}^n \to \mathbb{R}^{m \times m}$, let $f: \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}$ be a scalar field defined by $$f(x,y) := y^\top A(x) y$$ Can the gradient $\nabla_x f$ be written in closed-form as a function of $A(x)$, $x$ and $y$?
I tried to write the quadratic form as a sum $$\sum_i^m y_i \sum_j^m a_{ij}(x)y_j$$ Differentiation leads straight forward to the sum of vectors $\frac{\partial a_{ij}}{\partial x}$ instead of coefficients $a_{ij}(x)$, but I could not identify any structure.
$$ \nabla_{\bf x} f ({\bf x}, {\bf y}) = \begin{bmatrix} {\bf y}^\top \left( \partial_{x_1} {\bf A} ({\bf x}) \right) {\bf y} \\ \vdots \\ {\bf y}^\top \left( \partial_{x_n} {\bf A} ({\bf x}) \right) {\bf y} \end{bmatrix} $$