What is the result of
$$ \frac{\rm d}{{\rm d}t}\Big(A\left(x(t)\right)\cdot y\Big) $$
where variables $t\in\mathbb R$ and $y\in\mathbb R^n$, functions $x\colon=\mathbb R\to\mathbb R^\ell$ and $A\colon=\mathbb R^\ell\to\mathbb R^{m\times n}$.
Thanks in advance!
$$ \left(y\otimes I_{m\times m}\right)^{\mathsf T}\cdot D\cdot\dot x $$ where $D\in\mathbb R^{mn\times\ell}$ and the $i$-th column of $D$ is the derivative of ${\rm vec}(A)$ with respect to $x_i$ for $i=1,2,\dots,\ell$.
This answer is motivated by the accepted answer of the question What is the result of this type of derivative?