Product rule

Definition

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

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

Example

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

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

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

    $$=4e^{-x}\big(2x-x^2\big)$$

    $$=4xe^{-x}(2-x)$$

See also

• Chain rule