Differentiating Inverse Functions (DP IB Analysis & Approaches (AA)): Revision Note

Differentiating inverse functions

What is meant by an inverse function?

• Some functions are easier to handle with  (rather than ) as the subject

  • i.e.  in the form

  • This can be particularly true when dealing with inverse functions

• If  the inverse would be written as

  • Finding can be awkward

  • But a function and its inverse 'cancel' each other

    • This means

  • So you can write  instead

    • This expresses the same relationship between and as does

How do I differentiate inverse functions?

• With it is easier to differentiate “ with respect to ” rather than “ with respect to ”

  • I.e. to find rather than 

  • Note that will be in terms of 

• STEP 1
  For the function , the inverse will be

  Rewrite as

  • E.g. Find an expression in terms of for the derivative of , where

• STEP 2
  From find 

• STEP 3
  Find  using  (this will usually be in terms of )

• This can be used to find the gradient of the curve at a given point

  • Substitute the -coordinate of the point into your expression for

    • If the -coordinate is not given, you should be able to work it out from the original function and -coordinate

  • E.g. for the function used above, find the gradient of at the point where

    • You need to know the value of

      • 

    • Substitute into the expression for