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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Question
![**Problem Statement:**
Given \( f(x) = x^2 + 4 \), \( x \leq 0 \). Find \( (f^{-1})'(5) \).
**Options:**
- [ ] 10
- [ ] \(\frac{1}{2}\)
- [ ] \(-\frac{1}{2}\)
- [ ] 29
**Explanation:**
This problem asks for the derivative of the inverse function \( f^{-1} \) at the point 5. We are given the function \( f(x) = x^2 + 4 \) with the condition \( x \leq 0 \).
To solve the problem, utilize the formula for the derivative of an inverse function:
\[
(f^{-1})'(y) = \frac{1}{f'(x)}
\]
where \( y = f(x) \). In this case, solve \( x^2 + 4 = 5 \) for \( x \) within the given domain. Then, compute \( f'(x) \) to find the inverse's derivative.
**Method to Solve:**
1. Solve for \( x \) when \( x^2 + 4 = 5 \).
2. Differentiate \( f(x) \) to find \( f'(x) \).
3. Apply the inverse function derivative formula to determine \( (f^{-1})'(5) \).
This problem helps in understanding inverse functions and their derivatives, illustrating concepts in calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6c201a56-4b49-4358-9810-5f867d311d6a%2F7d26909a-3140-4c49-8f30-4e924370c728%2Fkm5g144_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Given \( f(x) = x^2 + 4 \), \( x \leq 0 \). Find \( (f^{-1})'(5) \).
**Options:**
- [ ] 10
- [ ] \(\frac{1}{2}\)
- [ ] \(-\frac{1}{2}\)
- [ ] 29
**Explanation:**
This problem asks for the derivative of the inverse function \( f^{-1} \) at the point 5. We are given the function \( f(x) = x^2 + 4 \) with the condition \( x \leq 0 \).
To solve the problem, utilize the formula for the derivative of an inverse function:
\[
(f^{-1})'(y) = \frac{1}{f'(x)}
\]
where \( y = f(x) \). In this case, solve \( x^2 + 4 = 5 \) for \( x \) within the given domain. Then, compute \( f'(x) \) to find the inverse's derivative.
**Method to Solve:**
1. Solve for \( x \) when \( x^2 + 4 = 5 \).
2. Differentiate \( f(x) \) to find \( f'(x) \).
3. Apply the inverse function derivative formula to determine \( (f^{-1})'(5) \).
This problem helps in understanding inverse functions and their derivatives, illustrating concepts in calculus.
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