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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![**Implicit Differentiation Exercise**
**Objective:**
Find \(\frac{dy}{dx}\) by implicit differentiation.
**Given Equation:**
\[ 8x^2 - 13xy + 5y^2 = 32 \]
**Instructions:**
1. Differentiate both sides of the equation with respect to \(x\).
2. Use the chain rule and the product rule where necessary.
**Steps:**
1. **Write the results of differentiating with respect to \(x\):**
The problem requires you to find the derivative of each term. Use:
- The product rule for the term \(-13xy\).
- The chain rule for the term \(5y^2\), treating \(y\) as a function of \(x\).
\[ \frac{d}{dx} (8x^2) - \frac{d}{dx} (13xy) + \frac{d}{dx} (5y^2) = 0 \]
2. **Complete the differentiation:**
\[
16x - (13y + 13x\frac{dy}{dx}) + 10y\frac{dy}{dx} = 0
\]
3. **Find \(\frac{dy}{dx}\):**
Solve the equation for \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = \boxed{}
\]
**Note:**
Fill in the details in the box provided on the answer sheet. This exercise involves algebraic manipulation to isolate \(\frac{dy}{dx}\) and express it in terms of \(x\) and \(y\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5350919d-d81b-4a95-8193-77894c363165%2Fcb97774f-7fcf-4eb1-928d-13a4be892e64%2F4rkkp5d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Implicit Differentiation Exercise**
**Objective:**
Find \(\frac{dy}{dx}\) by implicit differentiation.
**Given Equation:**
\[ 8x^2 - 13xy + 5y^2 = 32 \]
**Instructions:**
1. Differentiate both sides of the equation with respect to \(x\).
2. Use the chain rule and the product rule where necessary.
**Steps:**
1. **Write the results of differentiating with respect to \(x\):**
The problem requires you to find the derivative of each term. Use:
- The product rule for the term \(-13xy\).
- The chain rule for the term \(5y^2\), treating \(y\) as a function of \(x\).
\[ \frac{d}{dx} (8x^2) - \frac{d}{dx} (13xy) + \frac{d}{dx} (5y^2) = 0 \]
2. **Complete the differentiation:**
\[
16x - (13y + 13x\frac{dy}{dx}) + 10y\frac{dy}{dx} = 0
\]
3. **Find \(\frac{dy}{dx}\):**
Solve the equation for \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = \boxed{}
\]
**Note:**
Fill in the details in the box provided on the answer sheet. This exercise involves algebraic manipulation to isolate \(\frac{dy}{dx}\) and express it in terms of \(x\) and \(y\).
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