Given a function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ and a matrix $A \in \mathbb{R}^{n \times n}$. Is there a general formula for calculating the following derivative:
$$
\frac{\partial}{\partial x} f(x)^T A f(x) \tag{1} = ?
$$
I know that
$$
\frac{\partial}{\partial x} x^T A x = x^T(A + A^T) \overset{A = A^T}{=} 2 x^T A \tag{2}
$$
and the solution to $(1)$ will probably look similar to $(2)$, but I am stuck here since I am not sure how to apply the chain rule in the matrix case.
Edit: Regarding notation, we have
$$
\frac{\partial }{\partial x}f(x) = \begin{bmatrix} \frac{\partial}{\partial x_1} f_1(x) & \frac{\partial}{\partial x_2} f_1(x) & \cdots & \frac{\partial}{\partial x_n} f_1(x) \\
\frac{\partial}{\partial x_1} f_2(x) & \frac{\partial}{\partial x_2} f_2(x) & \cdots & \frac{\partial}{\partial x_n} f_2(x) \\
\vdots & \vdots & \ddots & \vdots \\
\frac{\partial}{\partial x_1} f_n(x) & \frac{\partial}{\partial x_2} f_n(x) & \cdots & \frac{\partial}{\partial x_n} f_n(x) \end{bmatrix}
$$
and
$$
x = \begin{bmatrix}
x_1 \\
x_2 \\
\vdots \\
x_n
\end{bmatrix} ,
f(x) = \begin{bmatrix}
f_1(x) \\
f_2(x) \\
\vdots \\
f_n(x)
\end{bmatrix}
$$
Best Answer
Given a differentiable vector field $\mathrm v : \mathbb R^n \to \mathbb R^n$ and a matrix $\mathrm A \in \mathbb R^{n \times n}$, let function $f : \mathbb R^n \to \mathbb R$ be defined by
$$f (\mathrm x) := \langle \mathrm v (\mathrm x), \mathrm A \mathrm v (\mathrm x) \rangle$$
whose directional derivative in the direction of $\mathrm y \in \mathbb R^n$ at $\mathrm x \in \mathbb R^n$ is
$$D_{\mathrm y} f (\mathrm x) := \lim_{h \to 0} \frac{f (\mathrm x + h \mathrm y) - f (\mathrm x)}{h} = \cdots = \langle \mathrm y, \mathrm J_{\mathrm v}^\top (\mathrm x) \, \mathrm A \, \mathrm v (\mathrm x) \rangle + \langle \mathrm J_{\mathrm v}^\top (\mathrm x) \, \mathrm A^\top \mathrm v (\mathrm x) , \mathrm y \rangle$$
where matrix $\mathrm J_{\mathrm v} (\mathrm x)$ is the Jacobian of vector field $\rm v$ at $\mathrm x \in \mathbb R^n$. Thus, the gradient of $f$ is
$$\nabla_{\mathrm x} f (\mathrm x) = \mathrm J_{\mathrm v}^\top (\mathrm x) \left( \mathrm A + \mathrm A^\top \right) \mathrm v (\mathrm x)$$