chain rule

Let f,g be differentiableMathworldPlanetmathPlanetmath, real-valued functions such that g is defined on an open set I, and f is defined on g(I). Then the derivativePlanetmathPlanetmath of the compositionMathworldPlanetmathPlanetmath fg is given by the chain ruleMathworldPlanetmath, which asserts that


The chain rule has a particularly suggestive appearance in terms of the Leibniz formalism. Suppose that z depends differentiably on y, and that y in turn depends differentiably on x. Then we have


The apparent cancellation of the dy term is at best a formal mnemonic, and does not constitute a rigorous proof of this result. Rather, the Leibniz format is well suited to the interpretationMathworldPlanetmathPlanetmath of the chain rule in terms of related rates. To wit:

The instantaneous rate of change of z relative to x is equal to the rate of change of z relative to y times the rate of change of y relative to x.

Title chain rule
Canonical name ChainRule
Date of creation 2013-03-22 12:26:43
Last modified on 2013-03-22 12:26:43
Owner matte (1858)
Last modified by matte (1858)
Numerical id 12
Author matte (1858)
Entry type Theorem
Classification msc 26A06
Related topic Derivative
Related topic ChainRuleSeveralVariables
Related topic ExampleOnSolvingAFunctionalEquation
Related topic GudermannianFunction