The derivative of det function I would like to compute $\frac{\partial}{\partial X} \det \left( A^H X A \right) $ where $A$ is a deterministic matrix. I get stuck and don't know how to start.
 A: I am not sure what the $H$ exponent is in your question, but assuming $A^H$ is a matrix of appropriate size, here's how I'd do it:
$$\frac{\partial{}}{\partial{X}}det(A^HXA)=\frac{\partial{}}{\partial{X}}det(A^H)det(X)det(A)=det(A^H)det(A)\frac{\partial{}}{\partial{X}}det(X)$$
The parts that do not depend on $X$ have been isolated. Now, by definition of the Jacobian (assuming $X\in{\mathbb{R}^{n+n}=\mathbb{R}^M}$)
$$\frac{\partial{det(X)}}{\partial{X}}=
\begin{pmatrix}
\frac{\partial{det(X)}}{\partial{x_1}}& ... & \frac{\partial{det(X)}}{\partial{x_M}} \\
\end{pmatrix}
$$ 
Now, each $\frac{\partial{det(X)}}{\partial{x_s}}$ has to be computed in a slightly different manner. To compute $\frac{\partial{det(X)}}{\partial{x_s}}$, where $x_s=X(i,j)$, One can use the definition of $det(X)$ as 
$$
det(X)=\sum_{j=1}^n(-1)^{i+j}X_{ij}det(X[i,j])
$$
Where $X[i,j]$ denotes the matrix resulting from suppressing the $i$-th row and $j$-th column of $X$. This yields 
$$
\frac{\partial{det(X)}}{\partial{X_{ij}}}=(-1)^{i+j}det(X[i,j])
$$
