Question Detail
Question
Differentiate the implicit functions x² siny - e2y cosx = 3x² + 2x - 5 with respect to x.
Correct Answer
Option C is the correct answer.
Detailed Explanation
Given the implicit equation: x² siny - e2y cosx = 3x² + 2x - 5.Differentiate both sides with respect to x using the product rule and chain rule:1. d/dx(x² siny) = 2x siny + x² cos y (dy/dx)2. d/dx(-e2y cosx) = -(d/dx(e2y) cosx + e2y d/dx(cosx)) = -(2e2y (dy/dx) cosx + e2y (-sinx)) = -2e2y cosx (dy/dx) + e2y sinx3. d/dx(3x² + 2x - 5) = 6x + 2Combine the derivatives:2x siny + x² cos y (dy/dx) - 2e2y cosx (dy/dx) + e2y sinx = 6x + 2Group terms with dy/dx:(x² cos y - 2e2y cosx) dy/dx = 6x + 2 - 2x siny - e2y sinxSolve for dy/dx:dy/dx = (6x + 2 - 2x siny - e2y sinx) / (x² cos y - 2e2y cosx)Comparing this with the given options, option C matches (assuming sinxe2y is equivalent to e2ysinx and the terms are rearranged).
Hint
Apply implicit differentiation. Remember to use the product rule for terms like x² siny and e2y cosx, and the chain rule for derivatives involving y (e.g., d/dx(siny) = cosy dy/dx).