Stelling

Als P(x) = f(x) · g(x) dan is P'(x) = f'(x) · g(x) + f(x) · g'(x).

Bewijs

Volgens de limietdefinitie van de afgeleide is

$P\text{'}(x)=\underset{h\to 0}{\lim }\frac{P(x+h)-P(x)}{h}=\underset{h\to 0}{\lim }\frac{f(x+h)\cdot g(x+h)-f(x)\cdot g(x)}{h}$

Met behulp van lineaire benadering van f(x + h) en g(x + h) wordt dit:

P'(x) = $\underset{h\to 0}{\lim }\frac{(f(x)+h\cdot f\text{'}(x))(g(x)+h\cdot g\text{'}(x))-f(x)\cdot g(x)}{h}$ =
        = $\underset{h\to 0}{\lim }\frac{h\cdot f\text{'}(x)\cdot g(x)+h\cdot f(x)\cdot g\text{'}(x)+h^2\cdot f\text{'}(x)\cdot g\text{'}(x)}{h}$

En dus is $P\text{'}(x)=\underset{h\to 0}{\lim }\left(f\text{'}(x)\cdot g(x)+f(x)\cdot g\text{'}(x)+h\cdot f\text{'}(x)\cdot g\text{'}(x)\right)=f\text{'}(x)\cdot g(x)+f(x)\cdot g\text{'}(x)$.