Chain Rule

Key Concepts

The chain rule is used to differentiate composite functions — where one function is applied inside another. Whenever you see a “function of a function”, you need the chain rule.

When to use it

Use the chain rule when differentiating expressions like:

• $(3x + 1)^5$ — a polynomial raised to a power
• $\sin(x^2)$ — trig of a non-trivial argument
• $e^{3x+1}$ — exponential of a non-trivial argument
• $\ln(x^2 + 1)$ — log of a non-trivial argument

The rule

If $y = f(g(x))$, let $u = g(x)$ so that $y = f(u)$. Then:

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

In function notation: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

Worked Examples

Example 1: Power of a linear expression

Differentiate $y = (2x + 3)^4$

Let $u = 2x + 3$, so $y = u^4$

$\frac{du}{dx} = 2, \qquad \frac{dy}{du} = 4u^3$

$\frac{dy}{dx} = 4u^3 \cdot 2 = 8(2x + 3)^3$

Example 2: Trig of a quadratic

Differentiate $y = \sin(x^2)$

Let $u = x^2$, so $y = \sin u$

$\frac{du}{dx} = 2x, \qquad \frac{dy}{du} = \cos u$

$\frac{dy}{dx} = \cos(x^2) \cdot 2x = 2x\cos(x^2)$

Example 3: Exponential with chain rule

Differentiate $y = e^{3x+1}$

Let $u = 3x + 1$, so $y = e^u$

$\frac{du}{dx} = 3, \qquad \frac{dy}{du} = e^u$

$\frac{dy}{dx} = 3e^{3x+1}$

Example 4: Natural log (A2)

Differentiate $y = \ln(x^2 + 1)$

Let $u = x^2 + 1$, so $y = \ln u$

$\frac{du}{dx} = 2x, \qquad \frac{dy}{du} = \frac{1}{u}$

$\frac{dy}{dx} = \frac{2x}{x^2 + 1}$

Example 5: Nested chain rule

Differentiate $y = e^{\sin(2x)}$

Apply the chain rule twice. Outer: $e^{(\cdot)}$, Middle: $\sin(\cdot)$, Inner: $2x$

$\frac{dy}{dx} = e^{\sin(2x)} \cdot \cos(2x) \cdot 2 = 2\cos(2x)\,e^{\sin(2x)}$

Common Patterns

These results follow directly from the chain rule and are worth memorising: