Product Rule

Example

Differentiate \(y=x^2(2x+1)^3\) with respect to \(x\).

Take \(u=x^2\) and \(v=(2x+1)^3\).

\(\dfrac{\mathrm{d}y}{\mathrm{d}x}\)\({}=x^2\bigl[3(2x+1)^2(2)\bigr]+(2x+1)^3(2x)\)\({}=6x^2(2x+1)^2+2x(2x+1)^3\)\({}=2x(2x+1)^2\bigl[3x+(2x+1)\bigr]\)\({}=2x(2x+1)^2(5x+1)\)

Why the rule works

If \(x\) increases by \(\delta x\), then \(u\) and \(v\) change by \(\delta u\) and \(\delta v\), and \((u+\delta u)(v+\delta v)-uv=u\,\delta v+v\,\delta u+\delta u\,\delta v\). Dividing by \(\delta x\) and letting \(\delta x\to0\), the last term vanishes, leaving \(u\dfrac{\mathrm dv}{\mathrm dx}+v\dfrac{\mathrm du}{\mathrm dx}\).