chain rule

Let $f,g$ be differentiable, real-valued functions such that $g$ is defined on an open set $I\subseteq ℝ$, 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

$${(f\circ g)}^{\prime }(x)=(f^{\prime }\circ g)(x)g^{\prime }(x),x\in I.$$

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

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

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 $x$.

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