Chain Rule

The chain rule differentiates a function inside another by multiplying the outer rate by the inner rate.

Identify structure → choose rule → verify. A formula or convergence question is inspected for structure and conditions, routed to an appropriate calculus rule or test, and checked against units, endpoints, differentiation, or the stated hypotheses.

When one quantity changes through another, their rates multiply.

Composite structure

A composite function has form

$$ y=f(g(x)). $$

The inner function $g$ acts first, and the outer function $f$ acts on its output. Differentiation must account for how fast both layers change.

The rule

The chain rule states

\frac{d}{dx}[f(g(x))]=f'(g(x))g'(x).

In Leibniz notation, if $y=f(u)$ and $u=g(x)$,

\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}.

The notation resembles cancellation and correctly tracks rates through the chain.

Worked example: a power of a polynomial

The factor $6x$ is essential; without it, only the outer layer has been differentiated.

Exponential and trigonometric examples

For $y=e^{4x-7}$,

y'=e^{4x-7}\cdot4.

For $y=\sin(x^2)$,

y'=\cos(x^2)\cdot2x.

The original inner expression remains inside the differentiated outer function.

Chain rule with product or quotient rules

Structure can require several rules. For

y=x^2\cos(3x),

use product rule outside and chain rule inside cosine:

y'=2x\cos(3x)-3x^2\sin(3x).

Marking the main operation first helps determine the outer rule.

Common mistakes

Differentiating only the outer function. Multiply by the inner derivative.

Changing the inner expression prematurely. Keep it intact inside $f'$.

Multiplying by the inner function rather than its derivative. Use $g'(x)$.

Missing layers in a nested composite. Trace from outside to inside.

Using chain rule instead of product rule for multiplied functions. Identify the main operation.