Question

F(x)=x^3 + x + 4
What is g'(6)?

Answer 1

f'(6) or
g'(6)

Answer 2

Sorry, I forgot to mention that g(x) is the inverse of f(x)

Answer 3

So yes, g'(6)

Answer 4

1/f'(x) tends to be defined as the inverse

Answer 5

so, whats our f'(x)?

Answer 6

3x^2 +1

Answer 7

yup thanks for that little piece of info, I just forgot that part, I think I might be able to solve it ^_^

Answer 8

and x=6 thats what; 36(3) + 1 = 90+18+1

Answer 9

yep, an inverse flips things about the y=x line so a straight line of a slope just flips as well

Answer 10

wait what?

Answer 11

I'm looking for g'(6)

Answer 12

right; and since g'(x) is the inverse of f'(x); g'(x) = 1/f'(x)

Answer 13

so far I have
1/(3x^2 + 1) = g(x)

Answer 14

so I find g'(x) from that and plugin 6?

Answer 15

you find g'(6) by finding 1/f'(6)

Answer 16

g'(6) = 1/f'(6)

if you know f'(6) then reciprocal it

Answer 17

ohhhh sorry I thought 1/f'(x) = g(x)

Answer 18

1/109?

Answer 19

think about it like this; a slope (y/x) IS a vector: <x,y>

when we flip things over the y=x line we get: <y,x> as the new slope

Answer 20

lol mind if we do this algebraically w/o a calculator? I'm taking a practice exam from section 1 so it would be cool if I could learn it like that

Answer 21

3(36) + 1 = 90 + 18 + 1 = 90+10+9 = 109

so yes, g'(6) = 1/109

Answer 22

Its weird, that's not one of the multiple choice answers, did I type something wrong?

Answer 23

maybe

Answer 24

Can you view the png attachment?

Answer 25

x^3 -> 3x^2
x -> 1
4 -> 0

f'(x) = 3x^2 + 1

the derivative is right; and the original f(x) doesnt undo very well to check its inverse with which is why this method is perfered

Answer 26

the inverse of a slope is simply swap top to bottom: y/x inverts to x/y

Answer 27

3(36) = 108 + 1 = 109 the flip of that is 1/109

let me see if im forgetting something with this tho ....

Answer 28

yea lol
i found this on paul's notes, maybe it will be helpful?
http://tutorial.math.lamar.edu/Classes/CalcI/DiffInvTrigFcns.aspx

Answer 29

yeah, lets do a problem I can test :)

y = x^2 ; y = sqrt(x)

y' = 2x ; y' = 1/2sqrt(x)

say x=4

y' = 8 ; y' = 1/4

hmmm, yeah, im missing something

Answer 30

f'(f(x)) eh

1/2(sqrt(x)) would be f'(f(x)) in this case

Answer 31

sqrt is square root right?

Answer 32

yeah

Answer 33

given that little side track :/

y = x^3+x+4

y-4 = x^3+x doesnt invert that well

Answer 34

http://oregonstate.edu/instruct/mth251/cq/Stage6/Lesson/inverseDeriv.html

heres what I was recalling tho

Answer 35

lol this is so confusing

Answer 36

http://www.wolframalpha.com/input/?i=invert+x%5E3%2Bx%2B4

now THAT would be a mess

Answer 37

im forgetting something pretty simple in all this ...

Answer 38

what is f(6)? by chance?

Answer 39

226

Answer 40

216+6+4 = 226

(6,226) the inverted point would then be (226,6)

Answer 41

so g(226) = f(6) ?

Answer 42

maybe

y=x^2 ; x=4, y=16

(4,16) inverts to (16,4)

2(4) = 8 ; but the inverse 1/2(sqrt(x)) needs to be 1/8 to make sense of this

ugh .... it bites getting old lol

Answer 43

hmm i have an idea

Answer 44

shouldnt we set the equation = to 6?

Answer 45

Solve for x
Then find the derivative of the function
and use it to finish solving for g'(6)

Answer 46

\[f(f^{-1}(x))=x\]
\[f(y)=x\]
\[f'(y)\ y'=1\]
\[y'=\frac{1}{f'(y)}\]
\[y'=\frac{1}{f'(f(x))}\]

Answer 47

yes, f(6) = 226

3(226)^2 + 1 tho is not going to get us one of those answers is it

Answer 48

Hmm...

Answer 49

y = f(x)
f(y) = x ; is our inverse

still comes out the same in the end doesnt it

Answer 50

comes out in the same end?

Answer 51

http://www.understandingcalculus.com/chapters/13/13-1.php

Answer 52

the inverse of a slope is its reciprocal; plain and simple.

if the slope at f(6) = 209; then the slope of its inverse is 1/209 ....

but either this is thinking askew or the problem you have is in error

Answer 53

x^2 + x + 4

2x + 1

13; 1/13 that aint it

-x^3 + x + 4

-3x^2 + 1

-107 ... not it

arghh!!

Answer 54

Hmm. These questions are annoying, I wish I had an answer key.
Hey well, thanks anyway! I'll move on to other practice problems and maybe I'll ask my teacher the next day.

Answer 55

ok, good luck with it all :)