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

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)

Inverses of Linear functions

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 -1 Slope is f Slope is m.

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)

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)

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!

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

f -1 In general, what does this tell us about the relationship between and ? (b, f -1(b)) f (a, f (a)) = (f -1(b), b) 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 (a, e a) f -1(x) = ln(x) (b, ln(b))