Derivative of Composite Functions — the Chain Rule

A composite function is a "function of a function", such as $(3x^2+5)^4$ (a power applied to a polynomial) or $\log(x^2+1)$ (a logarithm applied to a polynomial). Writing the outer function as $y = f(u)$ and the inner as $u = g(x)$, the chain rule states:

In words: differentiate the outer function (treating the inner as a single block), then multiply by the derivative of the inner function. The two $du$'s "cancel" as a memory aid, though the rule is a genuine theorem, not fraction cancellation.

Common special cases (with $u$ a function of $x$):

Method — chain rule step by step:

1. Identify the inner function $u = g(x)$ (usually what sits inside a bracket, root, power, log or trig function).
2. Differentiate the outer function with respect to $u$, leaving $u$ untouched inside.
3. Multiply by $\dfrac{du}{dx}$, the derivative of the inner function.
4. For deeply nested functions, apply the rule repeatedly, working from the outside in — one factor per layer.

Illustration. For $y = (x^2+1)^3$: the inner is $u = x^2+1$ with $\dfrac{du}{dx}=2x$; the outer is $u^3$ with $\dfrac{dy}{du}=3u^2$. So $\dfrac{dy}{dx}=3(x^2+1)^2 \cdot 2x = 6x(x^2+1)^2$.