Question

Challenge question:

Answer 1

\(\large g'(10) = f'^{-1}(10) \)

Answer 2

find f', set it equal to 10 and solve x

Answer 3

@zepdrix

Answer 4

hmmm

Answer 5

We can't actually calculate the inverse function.
So I'm thinking maybe we're suppose to use this formula:\[\Large \frac{d}{dx}f^{-1}(x)\quad=\quad \frac{1}{f'\left[f^{-1}(x)\right]}\]But I can't seem to make it work.
Hmm thinking...

Answer 6

@SithsAndGiggles @ranga

Answer 7

Yes, I would use the formula listed by @zepdrix above.

If g(x) is the inverse of f(x) then:
g'(10) = 1 / f'(g(10))
f(x) = ln(x-4) + 2x
f'(x) = 1/(x-4) + 2 to be evaluated at x = g(10)

We need to find g(10) first
g(x) being the inverse of f(x), What is g(10)?
is same as asking: What is the value of x that would make f(x) = 10
f(x) = ln(x-4) + 2x
10 = ln(x-4) + 2x
We can graph this or solve it by trial and error.
x = 5 will work.
g(10) = 5
f'(x) = 1/(x-4) + 2
f'(g(10)) = f'(5) = 1/(5-4) + 2 = 3
g'(10) = 1 / f'(g(10)) = 1/3