For differentiable matrix functions A,B:R→C n×n we have the product rule (AB)′ = AB′ + A′B . What is the difference to the case of plain functions?
closed as not a real question by Jacob Black, copper.hat, Dominic Michaelis, Thomas, Cameron Buie Mar 4 '13 at 18:34
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One big difference is that, for $n\geq 2$, you must watch and respect the order of the products.
Unlike with real valued functions where $$(fg)'=f'g+fg'=gf'+fg'=gf'+g'f=f'g+g'f.$$