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 $$ (f\circ g)'(x) = (f'\circ g)(x)\, g'(x), \quad x\in I. $$

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:

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 $x$