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 bythe 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 ofthe 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 formalmnemonic, and does not constitute a rigorous proof of this result.Rather, the Leibniz format is well suited to the interpretation of thechain rule in terms of related rates. To wit:

The instantaneous rate of change of $z$ relative to $x$ is equal to therate of change of $z$ relative to $y$ times the rate of change of$y$ relative to $x$.