Problem:
Let f(x) = x^3 and g(x) = x^3 - x for all x ∈ R. If f^(-1) denotes the inverse of the function f, then find the derivative of the composite function gof^(-1) at the point 2.

Solution:
To find the derivative of the composite function gof^(-1) at the point 2, we need to follow these steps:

Step 1: Find the inverse of the function f(x) = x^3
To find the inverse of f(x), we interchange x and y and solve for y:
x = y^3
Taking the cube root of both sides, we get:
y = ∛x
Therefore, the inverse of f(x) = x^3 is f^(-1)(x) = ∛x.

Step 2: Calculate the composite function gof^(-1)(x)
The composite function gof^(-1)(x) is obtained by substituting f^(-1)(x) = ∛x in place of x in the function g(x) = x^3 - x. So we have:
gof^(-1)(x) = (∛x)^3 - (∛x)
Simplifying this expression, we get:
gof^(-1)(x) = x - ∛x

Step 3: Find the derivative of gof^(-1)(x)
To find the derivative of gof^(-1)(x), we differentiate the expression x - ∛x with respect to x. The derivative of x is 1, and the derivative of ∛x can be found using the chain rule.
Using the chain rule, we have:
d/dx (∛x) = (1/3)(x^(-2/3))
Therefore, the derivative of gof^(-1)(x) is:
gof^(-1)'(x) = 1 - (1/3)(x^(-2/3))

Step 4: Evaluate the derivative at the point x = 2
To find the derivative of gof^(-1)(x) at the point x = 2, we substitute x = 2 into the derivative expression:
gof^(-1)'(2) = 1 - (1/3)(2^(-2/3))
Simplifying this expression, we get:
gof^(-1)'(2) = 1 - (1/3)(2^(-2/3))
gof^(-1)'(2) ≈ 0.694

Conclusion:
The derivative of the composite function gof^(-1) at the point 2 is approximately 0.694.