Question

Anyone good at inverses? Find g′(−1/5), where g(x) is the inverse of f(x)=x^9/x^2+4.

Answer 1

f(x)=x^9/x^2+4.
(is the same as) y=x^9/x^2+4
the inverse of that is x=y^9/y^2+4
g(-1/5) means insert -1/5 in place of x in g(x) the inverse function.
I hope that made sense.

Answer 2

as shown by ehuman , find the inverse of f(x) first, then plug in it -1/5

Answer 3

so is get -1/5=y^9/y^2+4 then what I do the derivative?

Answer 4

Just a minute. If g is the inverse of f then f is the inverse of g. So f=g^(-1)

Answer 5

I think they want you to use this fact
\[ (f^{-1})'(y_0) = \frac{1}{f'(x_0)} \]

Answer 6

you need to find x0
\[ \frac{x^9}{x^2+4}= \frac{1}{-5} \\
5x^9 +x^2+4=0
\]
by inspection, we (might) see x=-1 works

so now the problem is to find 1/f'(-1)

Answer 7

so should I plug in f(-1)?

Answer 8

you need to find the derivative of f(x)
that gives you f'(x)
then evaluate at x=-1
then "flip" because you want 1/f'(-1)

Answer 9

oh ok hold on one second let me see if i get right answer

Answer 10

I think I did something wrong I got 0

Answer 11

it's not zero.

what do you get for the derivative?

Answer 12

45x^8+2x?

Answer 13

is
\[ f(x) = \frac{x^9}{x^2+4} \] ?

Answer 14

9x^8/2x

Answer 15

you can use
\[ d \left(\frac{u}{v}\right) = \frac{v \ du - u \ dv}{v^2}\]

Answer 16

now I am confused were did u get that equation and what do I plug in for v and du?

Answer 17

it is the "quotient" rule... when you have to take the derivative of a fraction where x is in the top and bottom, you use it.

it says: bottom times the derivative of the top minus top times the derivative of the bottom

all divided by the bottom squared.

Answer 18

oh gotcha I get it now, give me a minute.

Answer 19

(x^2+4)(9x^8)-(x^9)(2x) and then do i plug in the -1?

Answer 20

yes, but remember all of that is divided by (x^2+4)^2

Answer 21

Oh yea lol, alright let me plug and see if I get answer.

Answer 22

I got 44.92

Answer 23

did you get 25 on the bottom?

Answer 24

the bottom is 25
the top is 43

so f'(-1)= 43/25

and 1/f'(-1) = 25/43 which is the answer

Answer 25

Hmm I hate 1.72 and typed it in and got it wrong and then I saw u flipped it but it still did not work.

Answer 26

I would enter is 25/43

Answer 27

Oh that worked, maybe it only takes fractions which makes sense for calculus! Thanks for all the help!