Derivative of a matrix multiplied by its conjugate transpose

Question

Let $A$ be a square matrix whose elements are complex. Is is true that the spatial derivative

$\frac{d}{dx}(AA^H) = AA^H_x + A_xA^H$, following the simple chain rule?

The superscript $H$ represents the conjugate transpose and subscript $x$ represents the derivative of that matrix.

edit: $A = A(x,y)$

Answer 1

Just a hint: Consider a fixed entry $ij$ in $AA^H$, which can be written as $$(AA^H)_{ij} = \sum_{k=1}^n a_{ik}(x)\overline{a_{jk}(x)}$$ and apply the standard differentiation here. Then proceed in a same manner for the right side of the equality.