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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![**Instruction: Differentiate.**
Given the function:
\[ y = \ln(4x^2 - 9x + 1) \]
You need to find the derivative \[ y' \].
**Explanation:**
To differentiate this function, apply the chain rule. The derivative of \(\ln(u)\) with respect to \(x\) is \(\frac{1}{u} \cdot \frac{du}{dx}\).
First, identify \(u = 4x^2 - 9x + 1\).
Calculate \(\frac{du}{dx}\) of \(u\):
\[ \frac{d}{dx}(4x^2 - 9x + 1) = 8x - 9 \]
Substitute these into the chain rule:
\[ y' = \frac{1}{4x^2 - 9x + 1} \cdot (8x - 9) \]
Simplify as needed to complete the differentiation. Fill in the expression for \( y' \) in the provided box.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2539764-cbb6-4f84-96a1-bc799fc276f3%2Fcd851c05-d64f-4f56-afc5-9a3ecda174c9%2F1nsmkt_processed.png&w=3840&q=75)
Transcribed Image Text:**Instruction: Differentiate.**
Given the function:
\[ y = \ln(4x^2 - 9x + 1) \]
You need to find the derivative \[ y' \].
**Explanation:**
To differentiate this function, apply the chain rule. The derivative of \(\ln(u)\) with respect to \(x\) is \(\frac{1}{u} \cdot \frac{du}{dx}\).
First, identify \(u = 4x^2 - 9x + 1\).
Calculate \(\frac{du}{dx}\) of \(u\):
\[ \frac{d}{dx}(4x^2 - 9x + 1) = 8x - 9 \]
Substitute these into the chain rule:
\[ y' = \frac{1}{4x^2 - 9x + 1} \cdot (8x - 9) \]
Simplify as needed to complete the differentiation. Fill in the expression for \( y' \) in the provided box.
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