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Find dy/dx by implicit differentiation. Then find the slope of the graph at the given point. tan(2x + y) = 2x, (0, 0) dy 2 2 cos (2n+y)-2 dx At (0, 0): y'= 0 =

Find dy/dx by implicit differentiation. Then find the slope of the graph at the given point. tan(2x + y) = 2x, (0, 0) dy 2 2 cos (2n+y)-2 dx At (0, 0): y'= 0 =

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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**Implicit Differentiation: Solving for \(\frac{dy}{dx}\) and Finding the Slope** In this lesson, we will solve for \(\frac{dy}{dx}\) by using implicit differentiation. Then, we will find the slope of the graph at a given point. **Problem:** \[ \tan(2x + y) = 2x, \quad (0, 0) \] **Required:** 1. Find \(\frac{dy}{dx}\) using implicit differentiation. 2. Evaluate \(\frac{dy}{dx}\) at the point \((0, 0)\). **Incorrect Solution:** \[ \frac{dy}{dx} = 2 \cos^2 (2\pi + y) - 2 \] *(This solution is marked incorrect.)* **Correct Solution:** Evaluate the derivative at the point \((0, 0)\): \[ \text{At } (0, 0): \quad y' = 0 \] *(This solution is marked correct.)* **Diagram:** There is no graph or diagram in this problem. The focus is solely on the algebraic manipulation and differentiation to find \(\frac{dy}{dx}\) and evaluate it at the given point. By correctly using implicit differentiation, you should be able to find the slope of the tangent line to the curve at the specific point \((0, 0)\).
Transcribed Image Text:**Implicit Differentiation: Solving for \(\frac{dy}{dx}\) and Finding the Slope** In this lesson, we will solve for \(\frac{dy}{dx}\) by using implicit differentiation. Then, we will find the slope of the graph at a given point. **Problem:** \[ \tan(2x + y) = 2x, \quad (0, 0) \] **Required:** 1. Find \(\frac{dy}{dx}\) using implicit differentiation. 2. Evaluate \(\frac{dy}{dx}\) at the point \((0, 0)\). **Incorrect Solution:** \[ \frac{dy}{dx} = 2 \cos^2 (2\pi + y) - 2 \] *(This solution is marked incorrect.)* **Correct Solution:** Evaluate the derivative at the point \((0, 0)\): \[ \text{At } (0, 0): \quad y' = 0 \] *(This solution is marked correct.)* **Diagram:** There is no graph or diagram in this problem. The focus is solely on the algebraic manipulation and differentiation to find \(\frac{dy}{dx}\) and evaluate it at the given point. By correctly using implicit differentiation, you should be able to find the slope of the tangent line to the curve at the specific point \((0, 0)\).
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