Scalar derivative of ${\rm tr}~[A(x)\log A(x)]$ where $A(x)$ is a square matrix How do i proceed to calculate
$$\frac{d}{dx}{\rm tr}\left[{A(x) \log A(x)}\right]$$
where $A(x) \in \mathbb{M}(n)$ and $x \in \mathbb{R}$? 
The $\log$ function is the one defined by the exponential map for matrices in the following sense: If $A=e^B$ then $B=\log A$, where $e ^X \equiv \sum_{k=0}^\infty X^k / k! $. The multiplication between $A$ and $\log A$ is matrix multiplication. 
Further assume that $A(x)$ are diagonalizable and nonsingular.
This problem arises in a statistical physics model where $A(x)$ is a density matrix depending on a scalar quantity and the trace expression is the (von Neumann) entropy. I tried to find it on the net but no luck and i got confused with the literature on matrix derivatives that are usually for derivation with respect to another matrix or with respect to a vector. Thanks anyone!

You just carry out the chain rule as you would normally.
$$ \frac{d}{dx}{\rm tr}\left[{A(x) \log A(x)}\right] = {\rm tr} [A'(x)\log A(x) + A(x)A^{-1}(x)A'(x)]$$
See here.
