#math/calculus For differentiable functions $f$ and $g$, the **product rule** is $ (fg)'(x)=f'(x)g(x)+f(x)g'(x). $ The derivative is not generally $f'g'$: both factors contribute to the product's change. To derive the rule, add and subtract $f(x+h)g(x)$ in the difference quotient: $ \begin{aligned} \frac{f(x+h)g(x+h)-f(x)g(x)}{h} &=f(x+h)\frac{g(x+h)-g(x)}{h}\\ &\quad+g(x)\frac{f(x+h)-f(x)}{h}. \end{aligned} $ As $h\to 0$, continuity of differentiable functions gives $f(x+h)\to f(x)$, so the limit is $f(x)g'(x)+g(x)f'(x)$. > [!Insight] > If a rectangle has side lengths $f$ and $g$, small changes give $\Delta(fg)=g\,\Delta f+f\,\Delta g+\Delta f\,\Delta g$. The final term is second-order small, leaving the product rule in the limit.