Question

Let g(x) be the inverse of f(x)=x^{3}+3x+5. Calculate g(9) [without finding a formula for g(x)] and then calculate g'(9).

g(9) = 1

g'(9) = ?

Answer 1

f(x)=y implies g(y)=x since g and f are inverses
so we need to find when f(x)=9

Answer 2

That is you need to solve x^3+3x+5=9
and you technically don't have to solve
just trial and error it

Answer 3

usually these are made so the guessing is easy

Answer 4

and it is pretty easy in this case

Answer 5

1 right?

Answer 6

1 is right
1+3+5=9

Answer 7

so f(1)=9 so f^(-1)(9)=1 (or what they called it g(9)=1)

Answer 8

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

Answer 9

\[\frac{d}{dx}g(x)|_{x=9}=\frac{1}{f'(g(9))}\]

Answer 10

So now find f'

Answer 11

and plug 1 into it

Answer 12

so \[f'=3x^2+3. \] plug in 1 you get 6 then plug that into 1/f'(g(9)) so the answer is 1/6. I think.

Answer 13

Sounds find.