Find the derivative of f(x) = (2/x-1)^3

Question

Find the derivative of f(x) = (2/x-1)^3

I need assistance in solving this problem, I am having trouble understanding what to do. I would greatly appreciate any help. I've been stuck on this problem for the past hour.

Answer 1

f(x) = [2 / (x-1)]³ = 8(x - 1)^-3

f'(x) = -24(x - 1)^-4 = -24 / (x - 1)⁴

Answer 2

for 2nd part

see below

for 1st part f(x) = (2/(x-1))^3 see the Mark M's post

f(x) = 2/(x-1)

use the quotient rule

denominator times derivative of numerator - numerator times derivative of denominator all over denominator squared

f'(x) = (x-1)(0) - 2(1) all over (x-1)^2 = -2/(x^2 -2x +1)

or

rewrite f(x) as 2(x-1)^-1 and apply the product rule

derivative = 1st term times derivative of 2nd term + 2nd term times derivative of 1st term

f'(x) =2(x-1)^-2 + (x-1)(0) = 2/(x-1)^2 = 2/(x^2-2x +1)

While I tend to go along with the other tutor, Mark M's response as correct, that depends on how you interpret the problem

As written you really wrote f(x) = ((2/x)-1)^3

so there's another answer, but you probably didn't intend to mean that,

but if, however unlikely, you did mean that, then

f(x) = (2/x -1)^2(2/x -1) = (4/x^2 - 4/x +1)(2/x -1) = 8/x^3 - 12/x^2 -6/x -1

and

f'(x) = -24/x^4 +24/x^3 +6/x^2