The product rule is a formula used to find the derivative of two functions multiplied together.
Given two differentiable functions $f(x)$ and $g(x)$, the derivative with respect to $x$ of the product $f(x)g(x)$ is given by:
\[\frac{\mathrm{d} }{\mathrm{d} x}\Bigl[f(x)g(x)\Bigr]=f'(x)g(x)+f(x)g'(x),\]
(A prime denotes the derivative with respect to $x$, i.e. $f'(x)=\dfrac{\mathrm{d} f}{\mathrm{d} x}$.)
Note: the product rule can be extended to compute the derivative of a product of more than two functions.
Example: Given three functions $f(x)$, $g(x)$ and $h(x)$, the derivative with respect to $x$ of the product $f(x)g(x)h(x)$ is:
\begin{align} \dfrac{\mathrm{d} }{\mathrm{d} x}\Bigl[f(x)g(x)h(x)\Bigr] &= f'(x)g(x)h(x)+f(x)\bigl(g(x)h(x)\bigr)' \\ &= f'(x)g(x)h(x)+f(x)g'(x)h(x)+f(x)g(x)h'(x) \end{align}
Given $f(x)=4x^3\sinh{x}$, find $\dfrac{\mathrm{d} f}{\mathrm{d} x}.$
Let $g(x)=4x^3$ and $h(x)=\sinh(x)$. Then by the product rule:
\begin{align} \dfrac{\mathrm{d} f}{\mathrm{d} x} &= g'(x)h(x)+g(x)h'(x) \\ &=\left(\dfrac{\mathrm{d} }{\mathrm{d} x}\left[4x^3\right]\right)\sinh{x}+4x^3\left(\dfrac{\mathrm{d} }{\mathrm{d} x}\left[\sinh{x}\right]\right) \\ \\ &=12x^2\sinh{x}+4x^3\cosh{x} \end{align}
Prof. Robin Johnson differentiates $x^2 \mathrm{e}^{\large{3x} }.$
Prof. Robin Johnson differentiates $(3x-2)^3(2x+1)^4.$
Test yourself: Numbas test on the product rule
Test yourself: Numbas test on differentiation, including the chain, product and quotient rules