Question

For each of the given functions f(x), find the derivative (f-¹1)'(c) at the given point c, first finding a = f-¹(c). (See Theorem 7, page 156 of the Stewart Essential Calculus textbook) 1. f(x) = 6x

Answer 1

The derivative of the inverse function (f^-1)'(c) for f(x) = 6x is found by first finding a = f^-1(c), then using the relationship (f^-1)'(c) = 1 / f'(a). For f(x) = 6x, this derivative is constant at 1 / 6.

Step-by-step explanation:

To find the derivative of the inverse function (f-1)'(c) at a given point c, we first identify a as f-1(c). Once a is known, we can use the fact that (f-1)'(c) = 1 / f'(a). Given f(x) = 6x, the derivative of f, which we represent as f', is constant at 6. Assuming we are given a specific c, we can subsequently find a such that f(a) = c. This implies that a = c / 6. Therefore, (f-1)'(c) = 1 / 6, and this derivative does not depend on the particular value of c since the derivative of f is a constant.