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

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

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.

f f -1 Slope is m. Slope is

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

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

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

f f -1 (a, f (a)) (b, f -1 (b)) In general, what does this tell us about the relationship between and ? Slope is.

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

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

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