$\frac{d(X'X)}{dX}=?$ Thanks a lot for reading my thread.
I am wondering what is the derivative of $X'X$ with respect to $X$?
Here $X$ is a vector/matrix, and $X'$ is the Hermitian matrix of $X$;
It would be great if you can also educate me with the real and complex case. 
$\frac{d(X'X)}{dX}=?$

This is just a notational issue because the object that you are differentiating and the differentiating variable are both matrices. You can define this in terms of a Kronecker product or vectorize the matrices before differentiating.  
To get some intuition, note that for a conformable vector $v$,
$$\frac{d (v'XX'v)}{dX}$$
is most naurally described as a matrix with the same dimensions as $X$ whose $(i,j)$ element is the derivative of the numerator with respect to the $(i,j)$ element of $X$.
A good reference is Magnus & Neudecker.
