Chain Rule (Edexcel A Level Maths: Pure): Revision Note

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 $\mathit{y}\mathbf{=}\mathbf{f}\mathbf{\left(}\mathbf{g}\mathbf{\right(}\mathit{x}\mathbf{\left)}\mathbf{\right)}$ then the chain rule can be written as:

$\frac{dy}{dx}=\mathrm{f}\text{'}\left(g\right(x\left)\right)\mathrm{g}\text{'}\left(x\right)$

  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:

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

•  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:

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