1. Derivatives of Inverse Functions
AP Calculus AB

2. Terminology If R = f(T) ... resistance is a function of temperature,
Then T = f -1(R) ... temperature is the inverse function of resistance.
f -1(R) is read "f-inverse of R“
is not an exponent
it does not mean reciprocal

3. Continuity and Differentiability
Given f(x) a function
Domain is an interval I
If f has an inverse function f -1(x) then …
If f(x) is continuous on its domain, then f -1(x) is continuous on its domain

4. Continuity and Differentiability
Furthermore …
If f(x) is differentiable at c and f '(c) ≠ 0 then f -1(x) is differentiable at f(c)
f(x)
f -1(x)
Note the counter example
f(x) not differentiable here
f -1(x) not differentiable here

5. Derivative of an Inverse Function
Given f(x) a function
Domain is an interval I
If f(x) has an inverse g(x) then g(x) is differentiable for any x where f '(g(x)) ≠ 0
And …
f '(g(x)) ≠ 0

6. We Gotta Try This! Given g(2) = 2.055 and So
Note that we did all this without actually taking the derivative of f -1(x)

7. Consider This Phenomenon
For (2.055, 2) belongs to f(x) (2, 2.055) belongs to g(x)
What is f '(2.055)?
How is it related to g'(2)?
By the definition they are reciprocals

8. Derivatives of Inverse Trig Functions
Note further patterns on page 177

9. Practice
Find the derivative of the following functions

10. More Practice
Given
Find the equation of the line tangent to this function at

11. Assignment
Lesson 3.6
Page 179
Exercises 1 – 49 EOO, 67, 69