Multivariate gaussian, MLE

How do you take the derivative of
\(\ln p(X|\mu,\Sigma)=-\frac{N D}{2}\ln(2\pi)-\frac{N}{2}\ln|\Sigma|-\frac{1}{2}\sum_{n=1}^N(x_n-\mu)^T\Sigma^{-1}(x_n-\mu)\)
with respect to \(\Sigma\)?
I'm really interested in what method you statisticians use when you have to deal with a novel expression (not something you can look up). Do you use matrix calculus? I'm aware of at least 3 approaches:
1) rewrite the expression in non-matrix form, compute the single partials and try to recompact the result into matrix form;
2) use Schonemann's method (paper);
3) use Magnus & Neudecker's method (book).
Just curious.


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I'm too lazy to follow those links but I just use matrix calculus. The first method you talk about is essentially what matrix calculus will do for you without doing it piece by piece. You just have to learn a few rules of operation and it's really not that bad.