# 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 $x$.

Title chain rule ChainRule 2013-03-22 12:26:43 2013-03-22 12:26:43 matte (1858) matte (1858) 12 matte (1858) Theorem msc 26A06 Derivative ChainRuleSeveralVariables ExampleOnSolvingAFunctionalEquation GudermannianFunction