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 Statement:**
Find \( y' \).
\( y = \sqrt{\ln(7x + 5)} \)
\[ y' = \boxed{} \]
---
**Solution Outline:**
To find the derivative \( y' \), we can use the chain rule. The function \( y \) is composed of an outer square root function and an inner natural logarithm function. We'll differentiate each component and combine them.
1. **Outer Function: \( u = \sqrt{v} \):**
- The derivative of \( u = \sqrt{v} = v^{1/2} \) is \( \frac{1}{2}v^{-1/2} = \frac{1}{2\sqrt{v}} \).
2. **Inner Function: \( v = \ln(7x + 5) \):**
- The derivative of \( v \) with respect to \( x \) is \( \frac{1}{7x + 5} \times 7 = \frac{7}{7x + 5} \).
3. **Combine Using the Chain Rule:**
- \( y' = \frac{1}{2\sqrt{\ln(7x + 5)}} \cdot \frac{7}{7x + 5} \).
After substituting back, simplify if necessary, and place your final expression in the box provided.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc7aab756-0f3b-421c-a400-c3f015f6942d%2F48519416-81d9-46e3-ad81-84e0604cca4e%2Fdd6mh6_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find \( y' \).
\( y = \sqrt{\ln(7x + 5)} \)
\[ y' = \boxed{} \]
---
**Solution Outline:**
To find the derivative \( y' \), we can use the chain rule. The function \( y \) is composed of an outer square root function and an inner natural logarithm function. We'll differentiate each component and combine them.
1. **Outer Function: \( u = \sqrt{v} \):**
- The derivative of \( u = \sqrt{v} = v^{1/2} \) is \( \frac{1}{2}v^{-1/2} = \frac{1}{2\sqrt{v}} \).
2. **Inner Function: \( v = \ln(7x + 5) \):**
- The derivative of \( v \) with respect to \( x \) is \( \frac{1}{7x + 5} \times 7 = \frac{7}{7x + 5} \).
3. **Combine Using the Chain Rule:**
- \( y' = \frac{1}{2\sqrt{\ln(7x + 5)}} \cdot \frac{7}{7x + 5} \).
After substituting back, simplify if necessary, and place your final expression in the box provided.
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