Derivative of $\det(XAX^H)$ wrt $X$?

Question

I found the following derivative in the Matrix cookbook https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf (equation 570): $$ \frac{\partial \det(XAX^H)}{\partial X} = \det(XAX^H)\left(AX^H(XAX^H)^{-1}\right)^T $$ is this correct or should we use Hermitian instead of Transpose?
I mean $$ \frac{\partial \det(XAX^H)}{\partial X} = \det(XAX^H)\left(AX^H(XAX^H)^{-1}\right)^H $$