$\frac{\partial}{\partial X^H} Tr ( X X^H X X ^H)$ Hi I know how to derive the following result below,
 \begin{eqnarray}
 \frac{\partial}{\partial X} Tr (X^\top A X B)= A^\top X B^\top +AXB\\
  \frac{\partial}{\partial X^\top} Tr (X^\top A X B)=\left( A^\top X B^\top +AXB\right)^\top
 \end{eqnarray}
where $X^\top$ is the real transpose of $X$.
 However I am trying to derive the following
 \begin{eqnarray}
 \frac{\partial}{\partial X^H} Tr ( X X^H X X ^H)=?
 \end{eqnarray}
 where $X^H$ is the complex (hermitian) transpose of the matrix $X$, and the real transpose is given by $X^\top$.  How can we approach this?  Thanks!
 A: Consider the function 
$$\eqalign{
 f(Y) &= \operatorname{tr}(AYAY) = A^T:YAY \cr
}$$where colon denotes the Frobenius (aka double-dot) product. 
Let's find the differential and gradient of this function
$$\eqalign{
df &= A^T:(dY\,AY+YA\,dY) \cr
  &= 2\,A^TY^TA^T:dY \cr
\cr
\frac{\partial f}{\partial Y} &= 2\,A^TY^TA^T \cr
\cr
}$$
To apply this to your question, set $\,Y\!=\!X^H\,\,$ and $\,\,A\!=\!X,\,\,$ yielding
$$\eqalign{
2\,X^TX^*X^T \cr
}$$

Update

(To address some of the questions in the comments)
Since 
$$\eqalign{
 \operatorname{tr}(AYAY) &= \operatorname{tr}(YAYA) \cr
}$$the differential and gradient wrt $A$ can be written down by interchanging $Y\leftrightarrow A$ in the previous result, i.e.
$$\eqalign{
df &= 2\,Y^TA^TY^T:dA \cr
\cr
\frac{\partial f}{\partial A} &= 2\,Y^TA^TY^T \cr
\cr
}$$
If you consider the function $f(A,Y)$, then its full differential is
$$\eqalign{
df &= 2\,Y^TA^TY^T:dA \,\,+\,\, 2\,A^TY^TA^T:dY \cr\cr
}$$
Rearrangements of the Frobenius product follow from its equivalence to the trace $$A:BC=\operatorname{tr}(A^TBC)$$ and the properties of the trace wrt transposing and/or cyclically shuffling its arguments.
So, for example, the following are all equal 
$$\eqalign{
 A:BC &= AC^T:B \cr
 &= B^TA:C \cr
 &= A^T:(BC)^T \cr
}$$
A: Write it down in terms of components, you get
$$\frac{\partial}{\partial \bar{X}_{ij}} \mathop{\rm tr}(X X^H X X^H)
=\frac{\partial}{\partial \bar{X}_{ij}} ( X_{kl} \bar{X}_{ml} X_{mn} \bar{X}_{kn}), $$
where for some components you need to compute $\frac{d\bar{z}}{dz}$ which doesn't exist, since the function $z\mapsto \bar{z}$ is not differentiable (said not holomorphic.)
