Chain Rule (Cambridge (CIE) AS Maths) : Revision Note

Chain Rule

What is the chain rule? 

• If y is a function of u, and u is a function of x, then the chain rule tells us that

• The chain rule allows us to differentiate more complicated expressions and composite functions

• You will often see and use the chain rule with different variables

  • This is particularly useful for connected rates of change

How do I differentiate (ax + b)ⁿ?

• For n = 2 you will most likely expand the brackets and differentiate each term separately

• If n > 2 this becomes time-consuming and if n is not a positive integer we need a different method completely

• The chain rule allows us to use substitution to differentiate any function in the form y = (ax + b)ⁿ

  • Let u = ax + b, then y = uⁿ

  • Differentiate both parts separately

    • and 

  • Put both parts into the chain rule

    • 

  • Substitute u = ax + b back into your answer

    • 

How do I differentiate √(ax+b)?

• The chain rule allows us to use substitution to differentiate any function in the form 

• Rewrite  

  • Let u = ax + b, then y = u^½

  • Differentiate both parts separately

    • and 

  • Put both parts into the chain rule

    • 

  • Substitute u = ax + b back into your answer

    • 

• This method can be used for any fractional power of any linear or non-linear expression

  • Provided you know how to differentiate the non-linear expression

How do I differentiate (f(x))ⁿ?

• This method can be used for any linear or non – linear expression

  • Let u = f(x) and follow the method above

  • In general if then   

• With practice you will be able to carry out this method without the need for u

  • This is essential for learning the reverse chain rule later in the course