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 \( y' \) for \( 4x^5 + 5y^4 - 1 = 132 \) at the point \( (2, -1) \).
**Solution:**
To find the derivative \( y' \), use implicit differentiation:
1. Differentiate both sides of the equation with respect to \( x \).
2. Solve for \( y' \).
**Equation:**
\[ 4x^5 + 5y^4 - 1 = 132 \]
**Differentiate:**
\[ \frac{d}{dx}(4x^5) + \frac{d}{dx}(5y^4) = \frac{d}{dx}(132) \]
**Simplified:**
\[ 20x^4 + 20y^3y' = 0 \]
Solve for \( y' \) at the point \( (2, -1) \):
**Substitute:**
\[ 20(2)^4 + 20(-1)^3y' = 0 \]
\[ 320 - 20y' = 0 \]
\[ -20y' = -320 \]
\[ y' = 16 \]
**Result:**
At \( (2, -1) \), \( y' = 16 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c4b35fb-d452-40ac-b8fa-a6c142b6dd62%2Fd206074d-23fc-49fe-955a-5ad1a2857914%2F1d8wpd_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem:**
Find \( y' \) for \( 4x^5 + 5y^4 - 1 = 132 \) at the point \( (2, -1) \).
**Solution:**
To find the derivative \( y' \), use implicit differentiation:
1. Differentiate both sides of the equation with respect to \( x \).
2. Solve for \( y' \).
**Equation:**
\[ 4x^5 + 5y^4 - 1 = 132 \]
**Differentiate:**
\[ \frac{d}{dx}(4x^5) + \frac{d}{dx}(5y^4) = \frac{d}{dx}(132) \]
**Simplified:**
\[ 20x^4 + 20y^3y' = 0 \]
Solve for \( y' \) at the point \( (2, -1) \):
**Substitute:**
\[ 20(2)^4 + 20(-1)^3y' = 0 \]
\[ 320 - 20y' = 0 \]
\[ -20y' = -320 \]
\[ y' = 16 \]
**Result:**
At \( (2, -1) \), \( y' = 16 \).
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