Question

f(x) = x/(x-9)
find (f^-1) ' (7/16)
(the derivative of the inverse function)

Answer 1

ok why is this wrong,
first i find the inverse
y= x/(x-9)
x = y/(y-9)
(y-9)x = y
yx-9x=y
yx-y = 9x
y(x-1)=9x
y = 9x/(x-1)
f^-1(x) = 9x/(x-1)

Answer 2

found the inverse, now i plug into the formula
\[1/(f'(f^-1(x))\]
So we find the derivative of this inverse, now we use quotient rule
(x-1)(9)-(9x)(1))/(x-1)^2
which simplifies down to -9/(x-1)^2
now the derivative of the inverse is 1/ all that
so
(x-1)^2/-9 is the derivative of the inverse
I plugged in 7/16 into the equation and got -9/256.
What did I do wrong???

Answer 3

@dude

Answer 4

@imqwerty @jimthompson5910 @jhonyy9

Answer 5

what is this math im scared to move on

Answer 6

hint:
If \(f(x) = \frac{x}{x-9}\), then \( f'(x) = -\frac{9}{(x-9)^2}\)

Answer 7

Yeah thats what I did, according to my formula
the derivative of an inverse function is
\[d/dx(f^-1(x)) = 1/f'(f^-1(x))\]

Answer 8

So since its 1 over that part
I just took (x-9)^2/-9 and plugged in 7/16 for x

Answer 9

You need to compute \(f^{-1}\left(\frac{7}{16}\right)\) first

\(f^{-1}(x) = \frac{9x}{x-1}\)

\(f^{-1}\left(\frac{7}{16}\right) = \frac{9\left(\frac{7}{16}\right)}{\frac{7}{16}-1}\)

\(f^{-1}\left(\frac{7}{16}\right) = m\)

The goal is to find \(\frac{1}{f'(m)}\)

Answer 10

does it not work the way I do it? where I find the inverse and then solve for the derivative of the inverse and lastly plug in the number? that's what my teacher taught so that's y i am confused.
He has an example
for a function f(x) = x^2
the inverse is \[f^-1(x) = \sqrt{x}\]
and then find derivative you get 1/2x^(-1/2)
and 1 / by that
so
you get 1/(2sqrt(x))

Answer 11

OMG me big bot. I get what you are saying now my b. I understood. Ty jimthompson

Answer 12

If you follow that method, then you won't need the formula \(\frac{d}{dx}[f^{-1}(x)] = \frac{1}{f'(f^{-1}(x))}\)

Answer 13

yep, i get it now thx jimthompson, i have like 2 more questions if u don't mind.

Answer 14

go ahead

Answer 15

yeah so what happened is my teacher taught 2 different methods, i did some weird thing and combined both of them idk why