A LevelMaths: PureAQARevision Notes7. Differentiation7.3 Further Differentiation7.3.3 Chain Rule

Chain Rule (AQA A Level Maths: Pure)

Revision Note

Author

Roger

Last updated

29 July 2023

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Chain Rule

What is the chain rule?

The chain rule is a formula that allows you to differentiate composite functions
If y is a function of u, and u is a function of x, then the chain rule tells us that:

\frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}

- Note that because u is a function of x, y is also a function of x

In function notation, if $y = f (g (x))$ then the chain rule can be written as:

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

This allows us to differentiate more complicated expressions:

Chain Rule Illustr 1_ex, A Level & AS Level Pure Maths Revision Notes

Chain Rule Illustr 2_ex, A Level & AS Level Pure Maths Revision Notes

What are the special cases of the chain rule?

It allows us to differentiate a function raised to a power:
- $\frac{d}{d x} ((f {(x)}^{n}) = n (f {(x))}^{n - 1} f' (x)$

Chain Rule Illustr 3_ex, AS & A Level Maths revision notes

It allows us to differentiate functions that are not given in the form y= f(x):
- $\frac{d y}{d x} = \frac{1}{\frac{d x}{d y}}$

Chain Rule Illustr 4_ex, AS & A Level Maths revision notes

There is a useful case which is used to integrate certain functions:
- $\frac{d}{d x} (\ln (f (x)) = \frac{f' (x)}{f (x)}$

Examiner Tip

The chain rule formulae are not in the exam formulae booklet – you have to know them.
When using the chain rule be sure to keep your functions straight (ie which function is y and which is u, or which is f and which is g).

Worked example

Chain Rule Example, A Level & AS Level Pure Maths Revision Notes

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Author: Roger

Expertise: Maths

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.