1. Derivatives of Inverse Functions
AP Calculus
Unit 6 Lesson 3
Mrs. Mongold

2. Continuity and Differentiability of Inverse Functions

3. The Derivative of An Inverse Function
Let f be a function that is differentiable on an interval I . If f has an inverse function g, then g is differentiable at any x for which f’(g(x)) does not equal 0. Moreover,
𝑔 ′ 𝑥 = 1 𝑓′(𝑔 𝑥 )

4. Example Let f(x) = ¼ x3 + x – 1
What is the value of f-1(x) when x = 3?
What is the value of (f-1)’(x) when x = 3?

5. Example Solution
We know an inverse exists. Use calculator to find out when f(x) = 3.

6. Example Solution
We know an inverse exists. Use calculator to find out when f(x) = 3.
We know that f(x) = 3 when x = 2 so we know that f-1(3) = 2

7. Example Solution
We know an inverse exists. Use calculator to find out when f(x) = 3.
We know that f(x) = 3 when x = 2 so we know that f-1(3) = 2
Because f is differentiable and has an inverse you can use 𝑔 ′ 𝑥 = 1 𝑓′(𝑔 𝑥 )

8. Example Solution
We know an inverse exists. Use calculator to find out when f(x) = 3.
We know that f(x) = 3 when x = 2 so we know that f-1(3) = 2
Because f is differentiable and has an inverse you can use 𝑔 ′ 𝑥 = 1 𝑓′(𝑔 𝑥 )
𝑓 −1 ′ 3 = 1 𝑓 ′ ( 𝑓 −1 3 ) = 1 𝑓 ′ (2)

9. Example Solution
We know an inverse exists. Use calculator to find out when f(x) = 3.
We know that f(x) = 3 when x = 2 so we know that f-1(3) = 2
Because f is differentiable and has an inverse you can use 𝑔 ′ 𝑥 = 1 𝑓′(𝑔 𝑥 )
𝑓 −1 ′ 3 = 1 𝑓 ′ ( 𝑓 −1 3 ) = 1 𝑓 ′ (2) =

10. Example Solution
We know an inverse exists. Use calculator to find out when f(x) = 3.
We know that f(x) = 3 when x = 2 so we know that f-1(3) = 2
Because f is differentiable and has an inverse you can use 𝑔 ′ 𝑥 = 1 𝑓′(𝑔 𝑥 )
𝑓 −1 ′ 3 = 1 𝑓 ′ ( 𝑓 −1 3 ) = 1 𝑓 ′ (2) = = 1 4

11. Homework
Verify f has an inverse then use the function and the given real number to find (f-1)(a)
1. f(x) = x3 – 1, a=26
2. f(x) = x3 + 2x – 1, a = 2
3. f(x) = sinx, −𝜋 2 ≤𝑥≤ 𝜋 2 , a = ½
4. f(x) = 𝑥+6 𝑥−2 , x>2, a = 3
5. f(x) = 𝑥−4 , 𝑎=2