chain rule
Let f,g be differentiable, real-valued functions such that g is defined on an open set I⊆ℝ, and f is defined on g(I). Then the derivative of the composition f∘g is given by the chain rule, which asserts that
(f∘g)′(x)=(f′∘g)(x)g′(x),x∈I. |
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
dzdx=dzdydydx. |
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 interpretation 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 |