Concept preview

What you will be able to do

1. 01Recognise when a function is a composite $f\bigl(g(x)\bigr)$, name the inner function $u=g(x)$ and the outer function $f(u)$, and state the chain rule $\dfrac{dy}{dx}=f'\bigl(g(x)\bigr)\cdot g'(x)$.
2. 02Differentiate composites built from powers, roots, and trigonometric functions — including nested compositions — by differentiating the outer function at the inner value and multiplying by the derivative of the inside.
3. 03Explain from the rates-of-change view why the chain rule multiplies derivatives, evaluate a composite derivative at a point, and avoid the most common error of dropping the inner derivative.

A composite function is a function inside a function. We write it as $y=f\bigl(g(x)\bigr)$, where $g$ is the inner function that acts first on $x$, and $f$ is the outer function that acts on the result. A clean way to see the structure is to introduce a helper letter for the inside: let $u=g(x)$, so that $y=f(u)$. For $y=\sqrt{x^2+1}$ the inner function is $u=x^2+1$ and the outer function is $y=\sqrt{u}$. For $y=\sin(3x)$ the inner is $u=3x$ and the outer is $y=\sin u$. Reading a formula and naming its inner and outer parts is the first and most important skill of this lesson — everything else follows from it.