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
4.5 2
![**Find the Derivative**
Given the function:
\[ y = \ln(3 - 5x) \]
Find the derivative \( y' \).
**Solution:**
To find the derivative of \( y = \ln(3 - 5x) \), we use the chain rule. The chain rule states that the derivative of \( \ln(u) \) is \( \frac{1}{u} \) times the derivative of \( u \). Here, \( u = 3 - 5x \).
1. Differentiate \( u = 3 - 5x \):
\[ \frac{du}{dx} = -5 \]
2. Apply the chain rule:
\[ y' = \frac{1}{3 - 5x} \cdot (-5) \]
\[ y' = \frac{-5}{3 - 5x} \]
Therefore, the derivative \( y' = \frac{-5}{3 - 5x} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F34d89729-f0b7-4660-b47f-8f1bb9a32e44%2Fb0a8d4a0-2f85-4fc1-99fc-1dba375a88a7%2F85tij7d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Find the Derivative**
Given the function:
\[ y = \ln(3 - 5x) \]
Find the derivative \( y' \).
**Solution:**
To find the derivative of \( y = \ln(3 - 5x) \), we use the chain rule. The chain rule states that the derivative of \( \ln(u) \) is \( \frac{1}{u} \) times the derivative of \( u \). Here, \( u = 3 - 5x \).
1. Differentiate \( u = 3 - 5x \):
\[ \frac{du}{dx} = -5 \]
2. Apply the chain rule:
\[ y' = \frac{1}{3 - 5x} \cdot (-5) \]
\[ y' = \frac{-5}{3 - 5x} \]
Therefore, the derivative \( y' = \frac{-5}{3 - 5x} \).
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