PlanetMath: product rule

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product rule

(Theorem)

The product rule states that if $f:\mathbb{R}\rightarrow\mathbb{R}$ and $g:\mathbb{R}\rightarrow\mathbb{R}$ are functions in one variable both differentiable at a point , then the derivative of the product of the two functions, denoted $f\cdot g$ , at is given by

$\displaystyle \frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}x}}\left(f\cdot g\right)(x_0) = f(x_0)g'(x_0) + f'(x_0)g(x_0).$

Proof

See the proof of the product rule.

Generalized Product Rule

More generally, for differentiable functions $f_1, f_2,\ldots,f_n$ in one variable, all differentiable at

, we have

$\displaystyle D(f_1\cdots f_n)(x_0)=\sum_{i=1}^n\left(f_i(x_0)\cdots f_{i-1}(x_0)\cdot Df_i(x_0)\cdot f_{i+1}(x_0)\cdots f_n(x_0)\right).$

Also see Leibniz' rule.

Example

The derivative of $x\ln\vert x\vert$ can be found by application of this rule. Let $f(x) = x, g(x) = \ln\vert x\vert$ , so that $f(x)g(x) = x\ln\vert x\vert$ . Then and $g'(x) = \frac{1}{x}$ . Therefore, by the product rule,

$\displaystyle \frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}x}}(x\ln\vert x\vert)$	$\displaystyle =$	$\displaystyle f(x)g'(x) + f'(x)g(x)$
	$\displaystyle =$	$\displaystyle \frac{x}{x} + 1\cdot\ln\vert x\vert$
	$\displaystyle =$	$\displaystyle \ln\vert x\vert + 1$

"product rule" is owned by mathcam. [ full author list (2) | owner history (1) ]

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Attachments:: proof of product rule (Proof) by mathcam
logarithmic proof of product rule (Proof) by Wkbj79

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Cross-references: differentiable functions, product, derivative, point, differentiable, variable, functions
There are 26 references to this entry.

This is version 9 of product rule, born on 2002-02-24, modified 2007-01-11.
Object id is 2628, canonical name is ProductRule.
Accessed 8126 times total.

Classification:

AMS MSC:

26A06 (Real functions :: Functions of one variable :: One-variable calculus)

Pending Errata and Addenda

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