Inverse Function Derivatives Relationship Explained
Understand the relationship between the derivatives of a function and its inverse, showcasing that they are reciprocals of each other at corresponding points. Explore how the slopes of f and f^-1 are related through visuals and linear function properties.
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Presentation Transcript
The derivatives of f and f -1 How are they related?
(4,6) Recall that if we have a one-to- one function f, we get f -1 from f, we switch every x and y coordinate. f -1 (2,3) (6,4) f (3,2) (-1,-1)
Inverses of Linear functions = + = ( ) f x mx b 1( ) x ??? f = + y mx b 1 m ( ) = y b x
Inverses of Linear functions 1 m ( ) = + = x b 1 ( ) f x mx b ( ) x f = + y mx b 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. 1 m ( ) = y b x
f -1 Slope is 1 m f Slope is m.
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)
What happens when we zoom in on these points? f -1 (2,3) f (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 ? f f ( ) 1 (b, f -1(b)) ( ) f Slope is . f a (a, f (a)) 1 ( ) f a ( ) = 1 ( ) b f
f -1 In general, what does this tell us about the relationship between and ? f f ( ) 1 (b, f -1(b)) f (a, f (a)) = (f -1(b), b) 1 ( ) f a ( ) = 1 ( ) b f But a = f -1(b), so . . . 1 ( ) = 1 ( ) b f 1 ( ( )) b f f
Upshot If f and f-1 are inverse functions, then their derivatives at corresponding points are reciprocals of one another : 1 ( ) = 1 ( ) x f 1 '( ( )) x f f
Derivative of the logarithm 1 ( ) f a 1 ea 1 e ( ) = 1 ( ) b f f (x) = e x = (a, e a) 1 b = = ln( ) f -1(x) = ln(x) b (b, ln(b)) 1 x d dx = ln( ) x
