1. The derivatives of f and f -1
How are they related?

2. Recall that if we have a one-to-one function f, we get f -1 from f, we switch every x and y coordinate.
(4,6)
f -1
(2,3)
(6,4) f
(3,2)
(-1,-1)

3. Inverses of Linear functions
In other words, the inverse of a linear function is a linear function and the slope of the function and its inverse are reciprocals of one another.

4. Inverses of Linear functions
In other words, the inverse of a linear function is a linear function and the slope of the function and its inverse are reciprocals of one another.

5. f -1
Slope is f
Slope is m.

6. What about the more general question
What about the more general question? What is the relationship between the slope of f and the slope of f -1?
f -1
(2,3) f
(3,2)

7. Note: the points where we should be comparing slopes are “corresponding” points. E.g. (3,2) and (2,3).
f -1
(2,3) f
(3,2)

8. f -1 f What happens when we “zoom in” on these points? (2,3) (3,2)
We see straight lines whose slopes are reciprocals of one another!

9. f -1
In general, what does this tell us about the relationship between and ?
(b, f -1(b)) f
(a, f (a))
But a = f -1(b), so . . .

10. Upshot
If f and f -1 are inverse functions, then their derivatives at “corresponding” points are reciprocals of one another :

11. Derivative of the logarithm
f (x) = e x
(a, e a)
f -1(x) = ln(x)
(b, ln(b))