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# chain rule

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

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

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:

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 $x$.

## Mathematics Subject Classification

26A06*no label found*

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