The Chain Rule

What you will learn

• Spot the inner and outer function.
• Differentiate composite functions.
• Apply the rule to powers of brackets.

Before you start: you should know the power rule and composite functions. See Derivatives.

The idea

The basic rules handle $x^5$ or $\sin x$, but what about $(3x + 1)^5$ or $\sin(x^2)$? These are composite functions: one function applied to the output of another. The chain rule tells you how to differentiate them.

Rates multiply

Imagine three gears. Gear A turns gear B twice as fast, and gear B turns gear C three times as fast. How fast does C turn compared with A? Six times. Rates of change chain together by multiplication.

In symbols, if $y$ depends on $u$ and $u$ depends on $x$:

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

It looks as if the $du$‘s cancel. They don’t literally, but it is a helpful memory aid.

Inner and outer

For $y = (3x + 1)^5$:

• the inner function is $u = 3x + 1$;
• the outer function is $y = u^5$.

The chain rule says: differentiate the outer function (leaving the inner one alone), then multiply by the derivative of the inner function.

Worked examples

Example 1: A power of a bracket

Differentiate $y = (3x + 1)^5$.

1. Let $u = 3x + 1$, so $y = u^5$.
2. $\dfrac{dy}{du} = 5u^4$ and $\dfrac{du}{dx} = 3$.
3. Multiply: $\dfrac{dy}{dx} = 5u^4 \times 3 = 15u^4$.
4. Substitute back: $15(3x + 1)^4$.

Answer: $\dfrac{dy}{dx} = 15(3x + 1)^4$

Example 2: A square root

Differentiate $y = \sqrt{x^2 + 9}$ and find the gradient at $x = 4$.

1. Rewrite as a power: $y = (x^2 + 9)^{1/2}$.
2. Outer: $\frac{1}{2}u^{-1/2}$. Inner derivative: $2x$.
3. Multiply: $\dfrac{dy}{dx} = \frac{1}{2}(x^2 + 9)^{-1/2} \cdot 2x = \dfrac{x}{\sqrt{x^2 + 9}}$.
4. At $x = 4$: $\dfrac{4}{\sqrt{25}} = \dfrac{4}{5}$.

Answer: $\dfrac{x}{\sqrt{x^2 + 9}}$; gradient $0.8$ at $x = 4$.

Example 3: Exponential growth

A population is modelled by $P = 500e^{0.2t}$. Find the rate of growth when $t = 5$.

1. Inner function $0.2t$ has derivative $0.2$.
2. $\dfrac{dP}{dt} = 500 \times 0.2\, e^{0.2t} = 100e^{0.2t}$.
3. At $t = 5$: $100e^{1} \approx 271.8$.

Answer: about $272$ individuals per unit time.

Example 4: A chain of three

Differentiate $y = \sin^2(4x)$, which means $[\sin(4x)]^2$.

1. Outermost: the square, giving $2\sin(4x)$.
2. Middle: $\sin$, whose derivative is $\cos(4x)$.
3. Innermost: $4x$, with derivative $4$.
4. Multiply all three: $2\sin(4x) \cdot \cos(4x) \cdot 4 = 8\sin(4x)\cos(4x)$.

Answer: $8\sin(4x)\cos(4x)$

The most common mistake is forgetting the final “times the derivative of the inside”. If the inside is anything more than plain $x$, the chain rule applies.

Why it matters

Real-world quantities are rarely simple functions of one variable; they are layered. The area of a ripple depends on its radius, which depends on time, so how fast is the area growing? The chain rule answers “related rates” questions like this. It is also the engine behind backpropagation, the algorithm that trains neural networks by passing derivatives back through layer after layer of composite functions.