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).
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.