So, let's say I have a scalar variable $x$, some matrix $D \in \mathbb{R}^{n \times n}$, and some constant vector $y \in \mathbb{R}^{n \times 1}$. We let $D$ be a function of $x$, i.e. $D = D(x)$.
Let there exist a function $F(x) = D(x)y$.
How do I take the derivative of $F(x)$ with respect to x?
More explicitly, $\cfrac{d}{dx}F(x) = $???
I just wrote it out explicitly for $n=2$, and came up with $\cfrac{d}{dx}F(x) = D'(x)y$.
You'll end up with $F'(x) = D'(x)y$, where $D'$ is the entrywise derivative of $D(x)$.