Chain Rule (Edexcel IAL Maths)

Spec point — 4.2

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:

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

• 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:

$\frac{dy}{dx}=f'(g(x))g'(x)$

This allows us to differentiate more complicated expressions:

What are the special cases of the chain rule?

• It allows us to differentiate a function raised to a power:

  • $\frac{d}{dx}((f(x)^{n})=n(f(x))^{n-1}f'(x)$

• It allows us to differentiate functions that are not given in the form y= f(x):

  • $\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}$

• There is a useful case which is used to integrate certain functions:

  • $\frac{d}{dx}(\mathrm{ln}(f(x))=\frac{f'(x)}{f(x)}$

Exam Hint

• 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