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



See the proof of the product rule (

0.1 Generalized Product Rule

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


Also see Leibniz’ rule (


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 and g(x)=1x. Therefore, by the product rule,

ddx(xln|x|) = f(x)g(x)+f(x)g(x)
= xx+1ln|x|
= ln|x|+1
Title product rule
Canonical name ProductRule
Date of creation 2013-03-22 12:27:57
Last modified on 2013-03-22 12:27:57
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 12
Author mathcam (2727)
Entry type Theorem
Classification msc 26A06
Related topic Derivative
Related topic ProofOfProductRule
Related topic ProductRule
Related topic PowerRule
Related topic ProofOfPowerRule
Related topic SumRule
Related topic ZeroesOfDerivativeOfComplexPolynomial