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 #34:**
Find \( y' \) if \( y = \frac{\cos(x-1)}{x-1} \).
---
**Solution Approach:**
To find the derivative \( y' \), use the quotient rule. For a function \( y = \frac{u(x)}{v(x)} \), the derivative is given by:
\[
y' = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}
\]
In this case:
- \( u(x) = \cos(x-1) \)
- \( v(x) = x-1 \)
Calculate \( u'(x) \) and \( v'(x) \):
- \( u'(x) = -\sin(x-1) \)
- \( v'(x) = 1 \)
Apply the quotient rule:
\[
y' = \frac{-\sin(x-1)(x-1) - \cos(x-1)(1)}{(x-1)^2}
\]
Simplify:
\[
y' = \frac{-(x-1)\sin(x-1) - \cos(x-1)}{(x-1)^2}
\]
This is the derivative \( y' \) using the quotient rule.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F63e2e494-b56f-474b-94e6-a1b0d865891f%2F0724fbde-10c4-4261-84b7-0b7413e789d7%2F502nzm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem #34:**
Find \( y' \) if \( y = \frac{\cos(x-1)}{x-1} \).
---
**Solution Approach:**
To find the derivative \( y' \), use the quotient rule. For a function \( y = \frac{u(x)}{v(x)} \), the derivative is given by:
\[
y' = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}
\]
In this case:
- \( u(x) = \cos(x-1) \)
- \( v(x) = x-1 \)
Calculate \( u'(x) \) and \( v'(x) \):
- \( u'(x) = -\sin(x-1) \)
- \( v'(x) = 1 \)
Apply the quotient rule:
\[
y' = \frac{-\sin(x-1)(x-1) - \cos(x-1)(1)}{(x-1)^2}
\]
Simplify:
\[
y' = \frac{-(x-1)\sin(x-1) - \cos(x-1)}{(x-1)^2}
\]
This is the derivative \( y' \) using the quotient rule.
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