Chain Rule (DP IB Applications & Interpretation (AI)): Revision Note

Written by: Lucy Kirkham

Reviewed by: Dan Finlay

Updated on 9 June 2025

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Chain Rule

What is the chain rule?

The chain rule states that if $y$ is a function of $u$ and $u$ is a function of $x$
- i.e. if $y = g (u)$ , where $u = f (x)$
- then
  $\frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}$
  - This is given in the formula booklet

In function notation this could be written

$y = g (f (x))$

$\frac{d y}{d x} = g^{'} (f (x)) f^{'} (x)$

How do I know when to use the chain rule?

Use the chain rule when you are trying to differentiate a composite function
- i.e. a “function of a function”
These can be identified as the variable (usually $x$ ) does not ‘appear alone’
- $\sin x$ is not a composite function, because $x$ ‘appears alone’
- $\sin (3 x + 2)$ is a composite function
  - $x$ is tripled and has 2 added to it before the sine function is applied

How do I use the chain rule?

STEP 1
Identify the two functions
Rewrite $y$ as a function of $u$ ; $y = g (u)$
Write $u$ as a function of $x$ ; $u = f (x)$
STEP 2
Differentiate $y$ with respect to $u$ to get $\frac{d y}{d u}$
Differentiate $u$ with respect to $x$ to get $\frac{d u}{d x}$
STEP 3
Obtain $\frac{d y}{d x}$ by applying the formula $\frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}$ and substituting $f (x)$ back in for $u$

Examiner Tips and Tricks

In trickier problems the chain rule may have to be applied more than once.

Are there any standard results for using chain rule?

There are five general results that can be useful

If...	then...
$y = {(f (x))}^{n}$	$\frac{d y}{d x} = n f^{'} (x) {(f (x))}^{n - 1}$
$y = e^{f (x)}$	$\frac{d y}{d x} = f^{'} (x) e^{f (x)}$
$y = \ln (f (x))$	$\frac{d y}{d x} = \frac{f^{'} (x)}{f (x)}$
$y = \sin (f (x))$	$\frac{d y}{d x} = f^{'} (x) \cos (f (x))$
$y = \cos (f (x))$	$\frac{d y}{d x} = - f^{'} (x) \sin (f (x))$

Examiner Tips and Tricks

You should aim to be able to spot and carry out the chain rule mentally (rather than writing out the substitution).

Every time you use it, you can say to yourself in your head “Differentiate the first function ignoring the second, then multiply by the derivative of the second function".

Worked Example

a) Find the derivative of $y = {(x^{2} - 5 x + 7)}^{7}$ .

b) Find the derivative of $y = \sin (e^{2 x})$ .

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Chain Rule (DP IB Applications & Interpretation (AI)): Revision Note

Chain Rule

What is the chain rule?

How do I know when to use the chain rule?

How do I use the chain rule?

Are there any standard results for using chain rule?

You've read 0 of your 5 free revision notes this week

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

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