Correction to f(x) and Reposted.....Derivative of the inverse of f(x) = a polynomial

Question

Find the derivative of f^-1(x) at x=6 if f(x) is defined by f(x) = x³ - x.

I found the formula g'(x) = 1 / f'(g(x)), where g(x) = f^-1(x).

 

Please explain in detail, because my teacher for AP Calculus AB has only shown us a procedure to find the result, while telling us not to pay attention to use of the formula. As I result, I don't understand these problems very well, although it seems like they are simple problems.

Answer 1

y = x^3 -x to find the inverse switch x and y and solve for the new y

x = y^3 -y, use the cubic formula which is not well known and very complicated

y = (cube root of (x/2 + sqr(x^2/4 + 1/27)+ (cube root of (x/2 - sqr(x^2/4 - 1/27)= inverse function

use the chain rule to find the derivative

"inverse" here does Not mean y =1/x