Chain Rule

When one quantity changes through another, their rates multiply.

Composite structure

A composite function has form

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

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

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

Outer-then-inner routine

1. Identify the outer function and leave its input unchanged.
2. Differentiate the outer function.
3. Multiply by the derivative of the inner function.
4. Repeat if more layers remain.

This avoids expanding expressions that were designed to stay composed.

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

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

The original inner expression remains inside the differentiated outer function.

Nested chains

For

there are three layers: square root, sum with a fourth power, and linear inner expression. Write powers:

Then

Each layer contributes one factor.

Chain rule with product or quotient rules

Structure can require several rules. For

use product rule outside and chain rule inside cosine:

Marking the main operation first helps determine the outer rule.

Rate interpretation

Suppose radius $r$ changes with time and area is $A=\pi r^2$. Then

Area changes through radius, so their rates multiply. Units confirm the chain: area per radius times radius per time gives area per time.

Reverse recognition

The chain rule explains why an expression such as

is the derivative of $\sin(x^2)$. Recognizing an inner derivative beside a composite function becomes important in integration.

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.

Quick self-check

• What is the outermost operation?
• What expression serves as its input?
• Did the differentiated outer function keep that input?
• Is the inner derivative multiplied afterward?
• Are there additional product, quotient, or chain layers?
• Do units or a numerical check support the result?

Related topics

• Function Composition
• Product Rule
• Quotient Rule
• Related Rates