The chain rule · Fathom

By the end of this lesson

By the end you can differentiate a bracket raised to a power, like $(3x - 1)^5$ or $\sqrt{2x^3 + 5}$ , in one line. This is called the chain rule.

One extra step

You already know how to differentiate $x^3$ : power down, take one off, so $3x^2$ .

When there's a bracket with $x$ inside it, think of what's inside the bracket as a box. Then:

Differentiate as normal: power down, take one off, and keep the box exactly the same.
Multiply by the derivative of the box.

That second step is the only new thing. That's the whole chain rule.

Why multiply? Think of $(3x - 1)^5$ . The box $3x - 1$ changes 3 times as fast as $x$ does.

So whatever rate the power gives you per unit of box, you get 3 times that per unit of $x$ .

It's like gears: if one gear turns 3 times as fast as the next, the speeds multiply. (The analogy only holds while the speeds stay fixed; for a curved box like $x^2 + 1$ the multiplier changes with $x$ .)

Definition

The chain rule

$\frac{d}{dx}\big(\,\boxed{\;\text{box}\;}\,\big)^n = n\big(\,\boxed{\;\text{box}\;}\,\big)^{n-1} \times \big(\text{derivative of the box}\big)$

The power comes down and drops by 1, like always.
The box doesn't change.
Then multiply by the derivative of the box.

Worked example: Now one you wouldn't want to expand

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

Have a go first?

Quick check

Write an expressionQuestion 1 · Recall

Multiple choiceQuestion 2 · Recall

Answer the 2 questions above to continue.