The traditional multivariate Gaussian density is given by:
$$p(x; \mu, \Sigma) := (2 \pi \det(\Sigma))^{-1/2} \exp(-\frac{1}{2} (x-\mu)^T (\Sigma)^{-1} (x-\mu)) $$
Suppose I reparameterize the density with another matrix $A$, such that $\Sigma := A^T A$:
$$p(x; \mu, A) := (2 \pi \det(A^T A))^{-1/2} \exp(-\frac{1}{2} (x-\mu)^T (A^T A)^{-1} (x-\mu)) $$
What is the matrix derivative of the log multivariate Gaussian PDF with respect to $A$? That is, what is
$$ \nabla_A \log p(x; \mu, A)$$