# Product Rule The *product rule* is a formula for the [[derivative]] of the *product of two functions*. >[!Example] Product Rule >If $h(x)=f(x)g(x)$, then >$h'(x)=f'(x)g(x)+f(x)g'(x)$ A geometric proof of the product rule using *areas* is possible. The size of $df$ multiplied by $dg$ is so small it can be *neglected*; its size is exaggerated below. By dividing the total change in area $df\;g(x)+dg\;f(x)$ by the differential $dx$, the product rule can be derived. ![[ProductRuleGeometric.svg]] >[!Proof by the limit definition]+ > Let $h(x)=f(x)g(x)$ and that $f$ and $g$ are both differentiable at $x$. > $\begin{align*} > h'(x)&=\lim_{\Delta x\rightarrow 0}\frac{h(x+\Delta x)-h(x)}{\Delta x} \\ > &=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)g(x+\Delta x)-f(x)g(x)}{\Delta x} \\ > &=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)g(x+\Delta x)-f(x)g(x+\Delta x)+f(x)g(x+\Delta x)-f(x)g(x)}{\Delta x} \\ > &=\lim_{\Delta x\rightarrow 0}\frac{[f(x+\Delta x)-f(x)]g(x+\Delta x)+f(x)[g(x+\Delta x)-g(x)]}{\Delta x} \\ > &=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}\lim_{\Delta x\rightarrow 0}g(x+\Delta x) \\ &\hphantom{=}\;+\lim_{\Delta x\rightarrow 0}f(x)\lim_{\Delta x\rightarrow 0}\frac{g(x+\Delta x)-g(x)}{\Delta x} \\ > \therefore h'(x)&=f'(x)g(x)+f(x)g'(x) > \end{align*}$ > [^1] [^1]: The limit definition typically uses $h$ to denote a difference in input. As $h(x)$ is in use, $\Delta x$ has replaced it instead.