The product rule is really the simplest rule for derivatives that apply to operations on numbers that gives something interesting. Other basic rules would be a

The product rule is really the simplest rule for derivatives that apply to operations on numbers that gives something interesting. Other basic rules would be addition: (f(x) + g(x))' = f'(x) + g'(x) This is just linearity (together with constant multiplication) And function composition (the chain rule): (f ∘ g)' = (f' ∘ g)⋅g' We would need to somehow figure out what should correspond to function composition. So if we want something that captures some important algebraic properties of derivatives the product rule would be a good place to look.