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
![**Topic: Implicit Differentiation Exercise**
**Problem:**
Find \(\frac{dy}{dx}\) using implicit differentiation.
Given:
\[ x \cdot \sin(y) = 1 \]
\[ x = 10 \]
**Solution Steps:**
1. Differentiate both sides of the equation \(x \cdot \sin(y) = 1\) with respect to \(x\):
\[ \frac{d}{dx}(x \cdot \sin(y)) = \frac{d}{dx}(1) \]
2. Use the product rule for the left side:
\[ \frac{d}{dx}(x \cdot \sin(y)) = \sin(y) + x \cdot \frac{d}{dx}(\sin(y)) \]
3. Apply the chain rule to \(\frac{d}{dx}(\sin(y))\):
\[ \frac{d}{dx}(\sin(y)) = \cos(y) \cdot \frac{dy}{dx} \]
4. Substitute back:
\[ \sin(y) + x \cdot \cos(y) \cdot \frac{dy}{dx} = 0 \]
5. Rearrange to solve for \(\frac{dy}{dx}\):
\[ x \cdot \cos(y) \cdot \frac{dy}{dx} = -\sin(y) \]
\[ \frac{dy}{dx} = -\frac{\sin(y)}{x \cdot \cos(y)} \]
6. Substitute \(x = 10\) to find the specific value:
\[ \frac{dy}{dx} = -\frac{\sin(y)}{10 \cdot \cos(y)} \]
(The specific solution depends on further information about \(y\).)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6b4bd3b-c23c-4e19-9ff7-b724dd0e50b1%2Fb77d05c7-25b9-4ae2-9703-700437d40332%2F1gb69u_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Implicit Differentiation Exercise**
**Problem:**
Find \(\frac{dy}{dx}\) using implicit differentiation.
Given:
\[ x \cdot \sin(y) = 1 \]
\[ x = 10 \]
**Solution Steps:**
1. Differentiate both sides of the equation \(x \cdot \sin(y) = 1\) with respect to \(x\):
\[ \frac{d}{dx}(x \cdot \sin(y)) = \frac{d}{dx}(1) \]
2. Use the product rule for the left side:
\[ \frac{d}{dx}(x \cdot \sin(y)) = \sin(y) + x \cdot \frac{d}{dx}(\sin(y)) \]
3. Apply the chain rule to \(\frac{d}{dx}(\sin(y))\):
\[ \frac{d}{dx}(\sin(y)) = \cos(y) \cdot \frac{dy}{dx} \]
4. Substitute back:
\[ \sin(y) + x \cdot \cos(y) \cdot \frac{dy}{dx} = 0 \]
5. Rearrange to solve for \(\frac{dy}{dx}\):
\[ x \cdot \cos(y) \cdot \frac{dy}{dx} = -\sin(y) \]
\[ \frac{dy}{dx} = -\frac{\sin(y)}{x \cdot \cos(y)} \]
6. Substitute \(x = 10\) to find the specific value:
\[ \frac{dy}{dx} = -\frac{\sin(y)}{10 \cdot \cos(y)} \]
(The specific solution depends on further information about \(y\).)
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