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**
Find the derivative of the function.
\[ y = \ln\left(\sqrt[9]{7x - 3}\right) \]
**Solution**
To find the derivative \( y' \), we need to use the chain rule and logarithmic differentiation.
Let's rewrite the function:
\[ y = \ln\left((7x - 3)^{1/9}\right) \]
Using the logarithmic identity \(\ln(a^b) = b \cdot \ln(a)\), we can simplify:
\[ y = \frac{1}{9} \ln(7x - 3) \]
Now, find the derivative \( y' \):
1. Use the constant multiple rule: if \( y = c \cdot f(x) \), then \( y' = c \cdot f'(x) \).
2. The derivative of \(\ln(7x - 3)\) is \(\frac{1}{7x - 3} \cdot (7)\) using the chain rule.
So,
\[ y' = \frac{1}{9} \cdot \frac{1}{7x - 3} \cdot 7 \]
\[ y' = \frac{7}{9(7x - 3)} \]
**Answer**
\[ y' = \frac{7}{9(7x - 3)} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F22096048-b12d-49d9-8219-de16642cbbb8%2F183a92f3-93f7-4ee8-a099-2b83644cda41%2Fzgr36t_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Find the derivative of the function.
\[ y = \ln\left(\sqrt[9]{7x - 3}\right) \]
**Solution**
To find the derivative \( y' \), we need to use the chain rule and logarithmic differentiation.
Let's rewrite the function:
\[ y = \ln\left((7x - 3)^{1/9}\right) \]
Using the logarithmic identity \(\ln(a^b) = b \cdot \ln(a)\), we can simplify:
\[ y = \frac{1}{9} \ln(7x - 3) \]
Now, find the derivative \( y' \):
1. Use the constant multiple rule: if \( y = c \cdot f(x) \), then \( y' = c \cdot f'(x) \).
2. The derivative of \(\ln(7x - 3)\) is \(\frac{1}{7x - 3} \cdot (7)\) using the chain rule.
So,
\[ y' = \frac{1}{9} \cdot \frac{1}{7x - 3} \cdot 7 \]
\[ y' = \frac{7}{9(7x - 3)} \]
**Answer**
\[ y' = \frac{7}{9(7x - 3)} \]
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