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单词 ProductRule
释义

product rule


The product ruleMathworldPlanetmath states that if f: and g: are functions in one variable both differentiableMathworldPlanetmathPlanetmath at a point x0, then the derivativeMathworldPlanetmath of the product of the two functions, denoted fg, at x0 is given by

ddx(fg)(x0)=f(x0)g(x0)+f(x0)g(x0).

Proof

See the proof of the product rule (http://planetmath.org/ProofOfProductRule).

0.1 Generalized Product Rule

More generally, for differentiable functions f1,f2,,fn in one variable, all differentiable at x0, we have

D(f1fn)(x0)=i=1n(fi(x0)fi-1(x0)Dfi(x0)fi+1(x0)fn(x0)).

Also see Leibniz’ rule (http://planetmath.org/LeibnizRule).

Example

The derivative of xln|x| can be found by application of this rule.Let f(x)=x,g(x)=ln|x|, so that f(x)g(x)=xln|x|. Then f(x)=1 andg(x)=1x. Therefore, by the product rule,

ddx(xln|x|)=f(x)g(x)+f(x)g(x)
=xx+1ln|x|
=ln|x|+1
Titleproduct rule
Canonical nameProductRule
Date of creation2013-03-22 12:27:57
Last modified on2013-03-22 12:27:57
Ownermathcam (2727)
Last modified bymathcam (2727)
Numerical id12
Authormathcam (2727)
Entry typeTheorem
Classificationmsc 26A06
Related topicDerivative
Related topicProofOfProductRule
Related topicProductRule
Related topicPowerRule
Related topicProofOfPowerRule
Related topicSumRule
Related topicZeroesOfDerivativeOfComplexPolynomial
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