Derivative Product Rule

The product rule gives the derivative of a product of functions.

Statement

If $f$ and $g$ are functions of $x$, then

$$\frac{d}{dx}(f\cdot g)=f\cdot \frac{d}{dx}g+\frac{d}{dx}f\cdot g$$

Derivation

The derivation of the product rule using the Limit Definition of the Derivative.

$$\begin{array}{rl}\frac{d}{dx}(f(x)\cdot g(x))& =\underset{h\to 0}{\lim }\frac{f(x+h)g(x+h)-f(x)g(x)}{h}\\ & =\underset{h\to 0}{\lim }\frac{f(x+h)g(x+h)-f(x)g(x+h)+f(x)g(x+h)-f(x)g(x)}{h}\\ & =\underset{h\to 0}{\lim }\left[\frac{(f(x+h)-f(x))g(x+h)}{h}+\frac{f(x)(g(x+h)-g(x))}{h}\right]\\ & =\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}\cdot \underset{h\to 0}{\lim }g(x+h)+f(x)\cdot \underset{h\to 0}{\lim }\frac{g(x+h)-g(x)}{h}\\ & =\frac{d}{dx}f(x)\cdot g(x)+f(x)\cdot \frac{d}{dx}g(x)\end{array}$$