Problem

(a) Show that $ f(x) = x + r^x $ is one-to-one. …

02:38

Problem 77 Hard Difficulty

(a) Suppose $ f $ is a one-to-one differentiable function and its inverse function $ f^{-1} $ is also differentiable. Use implicit differentiation to show that
$ (f^{-1})'(x) = \frac {1}{f'(f^{-1}(x))} $
provided that the denominator is not 0.
(b) If $ f(4) = 5 $ and $ f'(4) = \frac {2}{3}, $ find $ (f^{-1})'(5). $

Answer

(a) $\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)}$
(b) $$ \frac{3}{2} $$

More Answers

01:33

Frank L.

Topics

Derivatives

Differentiation

Book Cover for Calculus: Early Tra…

Calculus: Early Transcendentals

Chapter 3

Differentiation Rules

Section 5

Implicit Differentiation

Discussion

You must be signed in to discuss.

Top Calculus 1 / AB Educators

Catherine R.

Missouri State University

Anna Marie V.

Campbell University

Kayleah T.

Harvey Mudd College

Caleb E.

Baylor University

Watch More Solved Questions in Chapter 3

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

Problem 38

Problem 39

Problem 40

Problem 41

Problem 42

Problem 43

Problem 44

Problem 45

Problem 46

Problem 47

Problem 48

Problem 49

Problem 50

Problem 51

Problem 52

Problem 53

Problem 54

Problem 55

Problem 56

Problem 57

Problem 58

Problem 59

Problem 60

Problem 61

Problem 62

Problem 63

Problem 64

Problem 65

Problem 66

Problem 67

Problem 68

Problem 69

Problem 70

Problem 71

Problem 72

Problem 73

Problem 74

Problem 75

Problem 76

Problem 77

Problem 78

Problem 79

Problem 80

Video Transcript

in this public park every 1st 1 show that at the worst time back to be won over 100 years off. Ex found me know that off. Been worse off eggs. It's a function is one tour is equal X. He detected evidence off both sides with respect to X. You see that F prime off in search of X times. They're a joke in your function. Clams F in worse prime of X. Is it one from this three diesel issue that f commerce prime of eggs is equal to than one over f crime off a humorous off X. And that is same as what is given in the problem statement. All right, that was part of it in part B. Where has to find the worst time? Five. So we're gonna use the question that we just drive some F universe Primer five is equal to one over. Prime asked in Worse off fire. What? Has that been worse? A fire. Well, you know that effort for is it for if 405 it means that you're worse off 54 So then we have on over prior or four and primal forces, Given that history, he said. It is over to over three, so we find the answer to be then three or two.

Doruk I.

Topics

Derivatives

Differentiation

Book Cover for Calculus: Early Tra…

Calculus: Early Transcendentals

Chapter 3

Differentiation Rules

Section 5

Implicit Differentiation

Top Calculus 1 / AB Educators

Catherine R.

Missouri State University

Heather Z.

Oregon State University

Kayleah T.

Harvey Mudd College

Samuel H.

University of Nottingham