Product rule

Definition

Given two functions [ilmath]f:\mathbb{R}\rightarrow\mathbb{R} [/ilmath] and [ilmath]g:\mathbb{R}\rightarrow\mathbb{R} [/ilmath] which are differentiable (at [ilmath]p[/ilmath]) the composite function [ilmath]h:\mathbb{R}\rightarrow\mathbb{R} [/ilmath] where [ilmath]h=fg[/ilmath] has derivative:

• [math]\frac{dh}{dx}\Bigg|_p=\frac{d}{dx}[fg]\Bigg|_p=f(p)\frac{dg}{dx}\Bigg|_p+g(p)\frac{df}{dx}\Bigg|_p[/math]
• Phonetically first times derivative of second plus second times derivative of first

Example

• [math]4x^2e^{-x}[/math]
  • [math]\frac{d}{dx}\Big[4x^2e^{-x}\Big]=4x^2\frac{d}{dx}\Big[e^{-x}\Big]+e^{-x}\frac{d}{dx}\Big[4x^2\Big][/math]

    [math]=4x^2(-1)e^{-x}+4e^{-x}\frac{d}{dx}\Big[x^2\Big][/math]

    [math]=4e^{-x}\Big(\frac{d}{dx}\Big[x^2\Big]-x^2\Big)[/math]

    [math]=4e^{-x}\big(2x-x^2\big)[/math]

    [math]=4xe^{-x}(2-x)[/math]

See also

• Chain rule

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