Derivative of trace of inverse of a matrix function I am trying to derive the derivative of the trace of inverse of a matrix function (of X), i.e.
$$f(X)=Tr\left((HXH^{H}+I)^{-1}\right) $$
where $H\in R^{n\times m}, X\in R^{m\times m}$. 
So $HXH^{H}+I$ is invertible while $HXH^{H}$ is not. 
Any one know how to find $\frac{\partial f}{\partial X}$ ?
Thanks in advance!
 A: Note: In the below, I used $X$ instead of $H$ by accident.  Also, $X^*$ is my notation for $X^H$.

We can find this inverse by the chain rule.  Note that $f = f_1 \circ f_2 \circ f_3$, where
$$
f_1(A) = Tr(A)\\
f_2(A) = A^{-1}\\
f_3(A) = XAX^* + I
$$
The derivatives of these functions are given by
$$
[f_1'(A)](B) = Tr(B)\\
[f_2'(A)](B) = -A^{-1}BA^{-1}\\
[f_3'(A)](B) = XBX^*\\
$$
The chain rule tells us that
$$
[(f_1 \circ f_2 \circ f_3)'(A)] = \\
[f_1'(f_2(f_3(A)))]\circ
[f_2'(f_3(A))] \circ f_3'(A)
$$
So, all together, that gives us
$$
[f'(A)] = \\
(B \mapsto Tr[(XBX^* + I)^{-1}])\circ
(B \mapsto -A^{-1}(XBX^* + I)A^{-1})\circ
(B \mapsto XBX^*) =\\
(B \mapsto Tr[(XBX^* + I)^{-1}])\circ
(B \mapsto -A^{-1}(X[XBX^*]X^* + I)A^{-1}) =\\
(B \mapsto Tr[(XBX^* + I)^{-1}])\circ
(B \mapsto -A^{-1}(X^2BX^{*2} + I)A^{-1}) =\\
(B \mapsto Tr[(X[-A^{-1}(X^2BX^{*2} + I)A^{-1}]X^* + I)^{-1}])
$$
In other words, if $A$ is parameterized as a function of $t \in \Bbb R$, then we can write
$$
\frac{d}{dt}f(A(t)) = 
Tr[(X[-A^{-1}(X^2\frac{dA}{dt}X^{*2} + I)A^{-1}]X^* + I)^{-1}]
$$

Clarification of my derivatives:
By $[f'(A)](B)$, I mean that for every $A$, $f'(A)$ is a function on $B$.  You are right to be confused.  Let's look at an example.
Take $f(X) = Tr(X)$, and find its derivative my way.  In fact, $f$ is linear, so this will be easy: we have
$$
[f'(A)](B) = Tr(B)
$$
Note that, in this case, $A$ doesn't pop up in the definition of $f'$. In other words, $f$ has a constant derivative.  This makes sense because $f$ is linear.  Another way of writing the above is to say that if $A(t)$ is a matrix-valued function on $t \in \Bbb R$, then
$$
\frac{d}{dt}f(A(t)) = Tr \frac{dA}{dt}
$$
Now, my understanding of your notation is that
$$
\frac{\partial f}{\partial X}[i,j] = \frac{\partial f}{x_{ji}} 
$$
equivalently, define $A_{ij}(t) = A + t E_{ji}$ (where $E_{ji}$ is the matrix with $0$s everywhere except at the $j,i$ entry, where it has a $1$).  Then we should have
$$
\frac{\partial f}{\partial X}(A)[i,j] = 
\frac {d}{dt} f(A_{ij}(t))
$$
By my above statement, that means we can write
$$
\frac{\partial f}{\partial X}(A)[i,j] = Tr(E_{ji}) = 
\begin{cases}
1 & i=j\\
0 & i \neq j
\end{cases}
$$
In other words, $\frac {\partial f}{\partial X}$ is $I$, which is exactly what we expected.
A: I got an answer myself in a relative simple way: $$\frac{\partial f}{\partial X}=H^H((HXH^H+I)^{-2})^{H}H$$ through the chain rule. That is, let $U=HXH^H$ and $g(U)=Tr((U+I)^{-1})$, then $$\frac{\partial f}{\partial X_{i,j}}=Tr[(\frac{\partial g(U)}{\partial U})^H \frac{\partial U}{\partial X_{i,j}}]$$.Since we can easily get $\frac{\partial g(U)}{\partial U}=(U+I)^{-2}$ and 
$$U=\sum_i \sum_j X_{i,j} h_i h_j^H$$, we know $$\frac{\partial U}{\partial X_{i,j}}=h_i h_j^H$$.We thus have $$\frac{\partial f}{\partial X_{i,j}}=Tr[((HXH^H+I)^{-2})^{H}h_ih_j^{H}]=Tr[h_j^{H}((HXH^H+I)^{-2})^{H}h_i]=h_j^{H}((HXH^H+I)^{-2})^{H}h_i$$. 
Reorganizing them in a compact form we have the above result (hopefully there is no error). 
Actually the $X$ I am dealing with is diagonal, so the derivation is simpler than above. 
Thank you all for your attention and especially for @Omnomnomnom!
