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dy dx sin(xy) = y dy dx Find the = given the equation

dy dx sin(xy) = y dy dx Find the = given the equation

Calculus: Early Transcendentals
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 Statement:** Find the derivative \(\frac{dy}{dx}\) given the equation: \[ \sin(xy) = y \] **Solution:** \(\frac{dy}{dx} = \) [Provide your answer here based on the calculation using implicit differentiation or other techniques as applicable.] **Explanation of Concepts:** 1. **Implicit Differentiation:** - Use implicit differentiation when both \(x\) and \(y\) are mixed through a trigonometric function as shown in the equation \(\sin(xy) = y\). - Differentiate both sides of the equation with respect to \(x\), remembering to apply the chain rule for the \(\sin(xy)\) term. 2. **Chain Rule:** - When differentiating \(\sin(xy)\), recognize that \(xy\) itself is a product of two functions. Use the chain rule to differentiate effectively. 3. **Steps:** - Differentiate \(\sin(xy)\) using the chain rule to find \(\cos(xy) \cdot (x \cdot \frac{dy}{dx} + y)\). - Differentiate \(y\) on the right side as \(\frac{dy}{dx}\). - Solve for \(\frac{dy}{dx}\) by equating and simplifying the differentiated equation. Note: This explanation should guide users through solving a differentiation problem where the typical separation of variables might not be straightforward.
Transcribed Image Text:**Problem Statement:** Find the derivative \(\frac{dy}{dx}\) given the equation: \[ \sin(xy) = y \] **Solution:** \(\frac{dy}{dx} = \) [Provide your answer here based on the calculation using implicit differentiation or other techniques as applicable.] **Explanation of Concepts:** 1. **Implicit Differentiation:** - Use implicit differentiation when both \(x\) and \(y\) are mixed through a trigonometric function as shown in the equation \(\sin(xy) = y\). - Differentiate both sides of the equation with respect to \(x\), remembering to apply the chain rule for the \(\sin(xy)\) term. 2. **Chain Rule:** - When differentiating \(\sin(xy)\), recognize that \(xy\) itself is a product of two functions. Use the chain rule to differentiate effectively. 3. **Steps:** - Differentiate \(\sin(xy)\) using the chain rule to find \(\cos(xy) \cdot (x \cdot \frac{dy}{dx} + y)\). - Differentiate \(y\) on the right side as \(\frac{dy}{dx}\). - Solve for \(\frac{dy}{dx}\) by equating and simplifying the differentiated equation. Note: This explanation should guide users through solving a differentiation problem where the typical separation of variables might not be straightforward.
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