Question

Given that f(x) and its inverse, f^−1(x), are differentiable functions with f(x) and f'(x) values at x= 2, x = 4, and x = 6 as indicated in the table, find the derivative of f^−1(x) at x = 4. Show all your work.

Answer 1

½

Explanation:

f(4) = 2

(f^-1)(4) = 6

Because f(6) = 4

f((f^-1)(x)) = x

[f((f^-1)(x))]' = f'(f^-1(x)) × (f^-1(x))'

(x)' = f'(f^-1(x)) × (f^-1(x))'

1 = f'(f^-1(x)) × (f^-1(x))'

At x = 4

1 = f'(6) × (f^-1(4))'

1 = 2 × (f^-1(4))'

(f^-1(4))' = ½

Answer 2

1/2

Explanation:

By definition of inverse functions:

f(f⁻¹(x)) = x

Take derivative using chain rule:

f'(f⁻¹(x)) (f⁻¹(x))' = 1

(f⁻¹(x))' = 1 / f'(f⁻¹(x))

Evaluate at x=4:

(f⁻¹(4))' = 1 / f'(f⁻¹(4))

(f⁻¹(4))' = 1 / f'(6)

(f⁻¹(4))' = 1 / 2