### 14 - Product Rule

Let $y=f\left(x\right)g\left(x\right)$ , then

$\frac{dy}{dx}=\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x+h\right)g\left(x+h\right)-f\left(x\right)g\left(x\right)}{h}$

$=\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x+h\right)g\left(x+h\right)-f\left(x\right)g\left(x+h\right)+f\left(x\right)g\left(x+h\right)-f\left(x\right)g\left(x\right)}{h}$

$=\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x+h\right)g\left(x+h\right)-f\left(x\right)g\left(x+h\right)}{h}+\frac{f\left(x\right)g\left(x+h\right)-f\left(x\right)g\left(x\right)}{h}$

$=\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x+h\right)g\left(x+h\right)-f\left(x\right)g\left(x+h\right)}{h}+\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x\right)g\left(x+h\right)-f\left(x\right)g\left(x\right)}{h}$

$=\underset{h\to 0}{\mathrm{lim}}\frac{\left[f\left(x+h\right)-f\left(x\right)\right]g\left(x+h\right)}{h}+\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x\right)\left[g\left(x+h\right)-g\left(x\right)\right]}{h}$

$=\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x+h\right)-f\left(x\right)}{h}\underset{h\to 0}{\mathrm{lim}}g\left(x+h\right)+\underset{h\to 0}{\mathrm{lim}}f\left(x\right)\underset{h\to 0}{\mathrm{lim}}\frac{g\left(x+h\right)-g\left(x\right)}{h}$

$={f}^{\prime }\left(x\right)g\left(x\right)+f\left(x\right){g}^{\prime }\left(x\right)$