1. Derivatives of inverse functions

2. Inverse Functions Two functions are inverses if
The graph of f contains the point (a,b) if and only if the graph of the inverse contains (b,a)
Reflect over the line y=x

3. To Find Inverses Solve the equation for x Interchange x and y
Replace y with

4. let
a) sketch the graph
b) find the inverse
c) sketch the inverse

5. let
d) differentiate both f(x) and f-1(x).

6. let
find the slope of the graph of f(x) at (4, 2) and the slope of the inverse at (2, 4)
f) What conclusion can you make about these slopes?

7. Inverse Functions
At right are the graphs of a function f(x) and its inverse f-1(x).
Do you see a relationship between the slope of the graph of f at (a,b) and the slope of the graph of the inverse at (b,a)?

8. conclusion
Since slope m = dy/dx, it should make sense that switching x and y (for inverse functions) should produce reciprocal slopes for inverse functions

9. 2) Find the derivative of the inverse of

10. Derivative of an inverse & image points
If (a, b) is a point on f, then (b, a) is a point on ,
Given x = 5 on the inverse and y = 2, then (5,2) is on the inverse and (2,5) is the image point on the original. Therefore

11. 3) Find the derivative of the inverse

12. 4) Let and let be the inverse of f, then find

13. Let
and

14. 6) Find the derivative of the inverse

16. 8)

17. 9) Selected values of g(x) and g’(x) are given in the table. Find

18. Home Work Derivatives of Inverse functions Worksheet 5-E
Use a section header for each of the topics, so there is a clear transition to the audience.
Derivatives of Inverse functions Worksheet 5-E