CIE A Level Maths: Pure 1复习笔记5.1.4 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
  • • 

How do I differentiate √(ax+b)?

• 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
• 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