chain ruleChainRule2002-02-24 01:50:482004-09-27 14:57:11Theorem\usepackage{amsmath}
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\newtheorem{proposition}{Proposition}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 \emph{chain rule}, which asserts that
$$
(f\circ g)'(x) = (f'\circ g)(x)\, g'(x), \quad x\in I.
$$
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
$$
\frac{dz}{dx} = \frac{dz}{dy}\, \frac{dy}{dx}.
$$
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:
\begin{quote}
\em 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 $x$.
\end{quote}