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Suppose that f(x) = 2x + 2/7 is differentiable and has an inverse for æ > 0 and f(1) = 4. Find (f ')'(4). а) 1 b) O- 4 d) 2 e) O- 3

Suppose that f(x) = 2x + 2/7 is differentiable and has an inverse for æ > 0 and f(1) = 4. Find (f ')'(4). а) 1 b) O- 4 d) 2 e) O- 3

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Problem: Inverse Function Derivative** Suppose that \( f(x) = 2x + 2 \sqrt{x} \) is differentiable and has an inverse for \( x > 0 \) and \( f(1) = 4 \). Find \( \left( f^{-1} \right)'(4) \). **Options:** a) \( \frac{2}{3} \) b) \( -\frac{1}{4} \) c) \( \frac{1}{6} \) d) \( \frac{1}{3} \) e) \( -\frac{2}{3} \) **Explanation:** To find \( \left(f^{-1}\right)'(4) \), use the formula for the derivative of the inverse function: \[ \left(f^{-1}\right)'(y) = \frac{1}{f'(x)} \] where \( f(x) = y \). Given \( f(1) = 4 \), find \( f'(x) \) and evaluate it at \( x = 1 \). Then use the value to compute \( \left(f^{-1}\right)'(4) \).
Transcribed Image Text:**Problem: Inverse Function Derivative** Suppose that \( f(x) = 2x + 2 \sqrt{x} \) is differentiable and has an inverse for \( x > 0 \) and \( f(1) = 4 \). Find \( \left( f^{-1} \right)'(4) \). **Options:** a) \( \frac{2}{3} \) b) \( -\frac{1}{4} \) c) \( \frac{1}{6} \) d) \( \frac{1}{3} \) e) \( -\frac{2}{3} \) **Explanation:** To find \( \left(f^{-1}\right)'(4) \), use the formula for the derivative of the inverse function: \[ \left(f^{-1}\right)'(y) = \frac{1}{f'(x)} \] where \( f(x) = y \). Given \( f(1) = 4 \), find \( f'(x) \) and evaluate it at \( x = 1 \). Then use the value to compute \( \left(f^{-1}\right)'(4) \).
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