Suppose we have differentiable functions $f(x)$ and $g(x)$, and we want to find the derivative of their product $F(x)=f(x)g(x)$.

$$

\begin{align}

F'(x)&=\lim_{h \to 0}\frac{F(x+h)-F(x)}{h} \\

&= \lim_{h \to 0}\frac{f(x+h)g(x+h)-f(x)g(x)}{h} \\

&= \lim_{h \to 0} \frac{f(x+h)g(x+h)-f(x)g(x)+f(x+h)g(x)-f(x+h)g(x)}{h}\\

&= \lim_{h \to 0} \frac{f(x+h)g(x+h)-f(x+h)g(x)+f(x+h)g(x)-f(x)g(x)}{h} \\

&= \lim_{h \to 0} f(x+h)\left(\frac{g(x+h)-g(x)}{h}\right) + \lim_{h \to 0}g(x)\left(\frac{f(x+h)-f(x)}{h}\right) \\

&=f(x)g'(x) + g(x)f'(x)

\end{align}

$$

Extension

Using the Product Rule and the Chain Rule, can you derive the Quotient Rule for differentiation?