Question

For each of the given functions f(x), find the derivative (f^-1)'(c) at the given point c, first finding a=f^(-1)(c)
f(x)=4x+7x^9; c=-11
a=?

Answer 1

whatever you do, do not try to find the inverse of this function, that is not the point of the exercise

Answer 2

lol

Answer 3

you need only \(f^{-1}(-11)\) which you can do in your head

Answer 4

how does that help?

Answer 5

how do you find the inverse?

Answer 6

because once you have it, you know how to find
\[\frac{d}{dx}f^{-1}(-11)\]

Answer 7

you do ( i repeat, do not) find \(f^{-1}(x)\) but only \(f^{-1}(-11)\)

Answer 8

no no a thousand times no, you do not find \(f^{-1}(x)\) as an equation
it is unnecessary and too much work and might not even be possible

Answer 9

\[f^{-1}(-11)=-1\] by your eyeballs, because \(f(-1)=-11\)

Answer 10

im so lost.

Answer 11

satellite, she's doing the derivative of the inverse, not just the inverse
\[\frac{ df ^{-1} }{ dx } =\frac{ 1 }{\frac{ df }{ dx } } \]

Answer 12

therefore, since \(\frac{d}{dx}f^{-1}(x)=\frac{1}{f'(f^{-1}(x))}\) and since you know \(f^{-1}(-11)=-1\) your job is only to take the derivative of \(f\), evaluate it at -1 and take the reciprocal ,
that is all

Answer 13

@Algebraic! yes i know, it is the derivative of the inverse evaluated at a number

Answer 14

yeah.

Answer 15

it is this
\[\frac{d}{dx}f^{-1}(c)=\frac{1}{f'(f^{-1}(c))}\]

Answer 16

so we need only one number, namely \(f^{-1}(-11)=-1\) to do this problem

Answer 17

1/f'(-1)?

Answer 18

and therefore all you need is the derivative of \(f\) itself

Answer 19

4+63x^8

Answer 20

@monroe17 no, it is the reciprocal of the derivative of the original function, evaluated at the inverse function

Answer 21

ok so you have \(f'(x)=4+63x^8\) right? what you need is \(f'(-1)\)

Answer 22

you can do it with your eyes, i get \(f'(-1)=4+63=67\)

Answer 23

ah I see what @satellite73 is saying...

Answer 24

and so your answer to
\[\frac{d}{dx}f^{-1}(-11)\] is
\[\frac{1}{67}\] that is all
finito halas stick a fork in it

Answer 25

yeah, he's right, I misread the c= thing...

Answer 26

so can you simplify all that down into steps for me..?

Answer 27

again steps are
find \(f^{-1}(-11)=-1\)
take the derivative of \(f\) get \(f'(x)=4+63x^8\)
evaluate at \(x=-1\) get \(f'(-1)=67\)
take the reciprocal to get the derivative of the inverse evaluated at \(-11\) get
\[\frac{1}{67}\]

Answer 28

written in one line it is
\[\frac{d}{dx}f^{-1}(-11)=\frac{1}{f'(f^{-1}(-11))}=\frac{1}{f'(-1)}=\frac{1}{67}\]

Answer 29

the only hard part is finding \(f^{-1}(-11)\) but since \(f(x)=4x+7x^9\) the solution to
\[4x+7x^9=-11\] sort of jumps out at you

Answer 30

actually if you look at the way the question was written, it asked first for \(f^{-1}(-11)\) which is the correct first step. it called it \(a\)
you need to do that first

Answer 31

so for... x^2-13x+61 c=25..
= x^2-13x+61=25
f^(-1)(25)?

Answer 32

yeah you are trying to solve \(x^2-13x+61=25\)

Answer 33

x=9,4?

Answer 34

if you know it, then you know \(f^{-1}(25)\) although this is a quadratic function so it doesn't have an inverse unless you restrict the domain

Answer 35

on the interval [6.5, inf)

Answer 36

yeah see the problem? you are not sure which one, because the inverse in this case is not a well defined function
the quadratic is not one to one

Answer 37

oh well then it is one to one i suppose
pick 9

Answer 38

since 9 is the on in the interval
now look how easy the problem becomes now that you have it
the derivative is easy
evaluating it at 9 is easy
taking the reciprocal is real easy, you just write it

Answer 39

1/5 yay!

Answer 40

yay!

Answer 41

at first it looks confusing, then it is simple

Answer 42

do you know why it works?

Answer 43

why?

Answer 44

\[f(f^{-1}(x))=x\] by definition

Answer 45

take the derivative of both sides. left is easy right uses chain rule
\[f'(f^{-1}(x)\times (f^{-1}(x))'=1\] solve for \((f^{-1}(x))'\) get
\[(f^{-1}(x))'=\frac{1}{f'(f^{-1}(x))}\]

Answer 46

well i got my left and right backwards, but that is the idea
now you can use it to find the derivative of any inverse function, assuming you know the derivative of the original function

Answer 47

okay, thanks so much!