Saturday, December 26, 2015

Derivatives of Inverse Functions

This is my fourth year teaching calculus on some level.  Every year (until this one!) my students have really struggled with finding the derivative of inverse functions at a point, especially in the manner these questions are often phrased on AP Exams.

To me, they're some of the most straight-forward multiple choice questions the students encounter on the exam; yet, year after year they miss this question (at least on their unit tests and mock exam).

So, clearly, not as straight-forward as I thought...

This year I formalized a strategy for them in three steps.  Not all three of these steps are necessary every time; but, if my students took the time to follow all three steps, they got these questions correct.

Here are the steps:

If f and g are differentiable functions and g is the inverse of f, then to find g'(a):

  1. List all points given on f as ordered pairs.
  2. List the points you now know are on g (switch x and y).
  3. Follow this formula: g'(a)=1/(f'(g(a)).
Let me show you with a couple examples.  Here's a question I pulled from this website.


Following the steps, we would work this question as follows:


How about one that describes f as an algebraic or numeric function, such as this FRQ from 2007:


Students could certainly start with Step 1 again and work their way down, but I encourage them--once they get comfortable--to feel free to start with Step 2 and fill in the blanks as they see fit.  Here's how I would suggest they work this problem:


That's it!

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