Let $x \mapsto A(x)$ be a matrix-valued function and let $v, w$ be fixed vectors of compatible dimension such that the map $$v^T A(x) w$$ is well-defined and takes values in $\mathbb{R}$. What is the following derivative?
$$\frac{\partial v^TA(x)w}{\partial x}$$
I feel like it should be matrix-valued but I'm having trouble showing it. I've taken a look at the matrix-calculus cookbook and still no luck.
If I understand this correctly, your "matrix function" takes a real argument $x$ and gives a complex-valued matrix $A(x)$. Let $v, w$ be vectors such as you defined, and
$$f : \mathbb{R} \rightarrow \mathbb{C} \\ x \mapsto v^TA(x)w$$
Your question simply pertains to derivating $f$, so as you can see the result is simply a complex-valued function of a real argument.
Now, as to computing said derivative, writing $x \mapsto A(x)$ is actually hiding the fact that every cell of $A$ is expressed in terms of $x$, so derivating the above function in regards to $x$ (which by the way is a normal derivative and not a partial one) gives a matrix whose cells are the derivatives of the cells in regards to $x$, and :
$$f'(x) = v^TA'(x)w$$
which is still a complex-valued function of a real argument.