Chain Rule (CIE AS Maths: Pure 1)

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