Find d/dx f^-1 (4) where f(x) = 4 + 2x^3 + sin (πx/ 2) for −1 ≤ x ≤ 1.

Question

***The f^-1 (4) = f inverse (4)

Answer 1

f(x) = 4 + 2x³ + sin(πx/2) f'(x) = 6x² + (π/2)cos(πx/2)

f(0) = 4, so f^-1(4) = 0

d/dx(f^-1(4)) = 1 / [d/dx(f(0)] = 1/(π/2) = 2/π

Note: The derivative of f^-1(x) at the point (a,b) is the reciprocal of the derivative of f(x) at (b, a).