PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (211)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (36)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Low Entry average rating: Very high
chain rule (Theorem)

Let $ f, g$ be differentiable, real-valued functions such that $ g$ is defined on an open set $ I\subseteq \mathbb{R}$, and $ f$ is defined on $ g(I)$. Then the derivative of the composition $ f\circ g$ is given by the chain rule, which asserts that

$\displaystyle (f\circ g)'(x) = (f'\circ g)(x)\, g'(x), \quad x\in 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

$\displaystyle \frac{dz}{dx} = \frac{dz}{dy}\, \frac{dy}{dx}. $
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$.



"chain rule" is owned by matte. [ full author list (2) | owner history (1) ]
(view preamble)

View style:

See Also: derivative, chain rule (several variables)


Attachments:
chain rule (several variables) (Theorem) by rmilson
example of chain rule (Example) by rmilson
proof of chain rule (Proof) by n3o
Log in to rate this entry.
(view current ratings)

Cross-references: related rates, interpretation, mnemonic, terms, composition, derivative, open set, functions, differentiable
There are 45 references to this entry.

This is version 9 of chain rule, born on 2002-02-24, modified 2004-09-27.
Object id is 2561, canonical name is ChainRule.
Accessed 12478 times total.

Classification:
AMS MSC: 26A06 (Real functions :: Functions of one variable :: One-variable calculus)
Pending Errata and Addenda
None.
[ View all 3 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)