The Chain Rule: Differentiate Composite Functions

The Chain Rule applies when one function is evaluated inside another: if $g$ is differentiable at $x$ and $f$ is differentiable at $g(x)$, then . The lesson makes that extra inner-derivative factor visible by comparing a tempting shortcut with an expanded polynomial check. 1

Three complete examples cover a polynomial power, a trigonometric composite, and nested layers. A side-by-side comparison of and shows why identifying the outside operation comes first, followed by a compact exam workflow for preserving every factor and earning method credit.