The chain rule in calculus

Calculus is hard. In most textbooks and calculus classes, the chain rule (f∘g)′(x) = f′(g(x))∘g′(x) is either not proved, or only partially proved. The reason is that the proof requires knowledge of topology not covered in the first two years of university, and most importantly, the result fails for infinite dimensional normed spaces even if the functions are assumed to be continuous with nonzero derivative. I will present an explicit counterexample.