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:**
Find \( y' \) if \( y = \ln(7x^2 + 3y^2) \).
**Solution:**
To find the derivative \( y' \), we need to apply implicit differentiation to the given function \( y = \ln(7x^2 + 3y^2) \).
**Steps:**
1. Differentiate both sides with respect to \( x \):
\[
\frac{d}{dx}[y] = \frac{d}{dx}[\ln(7x^2 + 3y^2)]
\]
2. The derivative of \( y \) with respect to \( x \) is \( y' \).
3. For the right side, use the chain rule:
\[
\frac{1}{7x^2 + 3y^2} \cdot (14x + 6yy')
\]
4. Equate and solve for \( y' \):
\[
y' = \frac{14x + 6yy'}{7x^2 + 3y^2}
\]
**Final expression for \( y' \):**
Organize the terms and solve for \( y' \) to find the complete expression.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4c444b80-0f04-4660-83de-5c28618d5488%2F5bcfe873-1161-48e0-b497-69bd19f44dea%2Fd8fbbl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem:**
Find \( y' \) if \( y = \ln(7x^2 + 3y^2) \).
**Solution:**
To find the derivative \( y' \), we need to apply implicit differentiation to the given function \( y = \ln(7x^2 + 3y^2) \).
**Steps:**
1. Differentiate both sides with respect to \( x \):
\[
\frac{d}{dx}[y] = \frac{d}{dx}[\ln(7x^2 + 3y^2)]
\]
2. The derivative of \( y \) with respect to \( x \) is \( y' \).
3. For the right side, use the chain rule:
\[
\frac{1}{7x^2 + 3y^2} \cdot (14x + 6yy')
\]
4. Equate and solve for \( y' \):
\[
y' = \frac{14x + 6yy'}{7x^2 + 3y^2}
\]
**Final expression for \( y' \):**
Organize the terms and solve for \( y' \) to find the complete expression.
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