I want to take the derivate of a rather complicated matrix expression. Is it possible to do this in MATLAB or Wolfram Alpha? Here is what I am trying to calculate:

\begin{equation} \frac{\partial}{ \partial {\bf X}} {\bf a}^T({\bf I}+\alpha {\bf X}^T{\bf A}^T{\bf A}{\bf X})^{-1}{\bf a} \end{equation}

Or if you know a cookbook where I can find this derivative, it would be highly appreciated! Thank you.