Product Rule

Reference > Calculus: Differentiation

Description
\[(fg)'=f'g+fg'\]
Examples
\[\frac{d}{dx} \sin{x}{x}^{2}\]
1
Regroup terms
\[\frac{d}{dx} {x}^{2}\sin{x}\]

2
Use the Product Rule to find the derivative of \({x}^{2}\sin{x}\)
The product rule states that \((fg)'=f'g + fg'\)
\[(\frac{d}{dx} {x}^{2})\sin{x}+{x}^{2}(\frac{d}{dx} \sin{x})\]

3
Apply the Power Rule: \(\frac{d}{dx} {x}^{n}=n{x}^{n-1}\)
\[2x\sin{x}+{x}^{2}(\frac{d}{dx} \sin{x})\]

4
The derivative of \(\sin{x}\) is \(\cos{x}\)
\[2x\sin{x}+{x}^{2}\cos{x}\]

Done