The derivative of the determinant of a Kronecker product

For an invertible matrix $A$, we have the identity \begin{align} \dfrac{\partial \det A}{\partial A} = \det A (A^{-1})^T \end{align} where the $T$ denotes the transpose operation.

How does this formula change when considering the Kronecker product of $A$ with some other (invertible) matrix $B$? For example, how does one compute the following:

\begin{align} \dfrac{\partial \det \left[A\otimes B\right]}{\partial A} \end{align}

$A$ being ${n\times n}$ matrix and $B$ being ${p\times p}$ matrix. We know that $$\det[A\otimes B]=(\det{A})^p(\det{B})^n$$ (Notice how the power switches between $n$ and $p$)
Then the derivative is $$\frac{\partial(\det{A})^p(\det{B})^n}{\partial A}=(\det{B})^np(\det A)^{p-1}\det A(A^{-1})^T=p(\det B)^n(\det A)^p(A^{-1})^T$$
$$\frac{\partial\det[A\otimes B]}{\partial A}=p\det[A\otimes B](A^{-1})^T$$