Question Detail
Question
Differentiate the implicit functions x2 sin y - e2y cos x = 3x2 + 2x - 5 with respect to x. Find dy/dx.
Correct Answer
Option C is the correct answer.
Detailed Explanation
Differentiate each term with respect to x:1. d/dx(x2 sin y) = 2x sin y + x2 cos y (dy/dx) (using product rule)2. d/dx(-e2y cos x) = -(d/dx(e2y) cos x + e2y d/dx(cos x)) = -(2e2y (dy/dx) cos x + e2y (-sin x)) = -2 cos x e2y (dy/dx) + sin x e2y3. d/dx(3x2 + 2x - 5) = 6x + 2Combine these:2x sin y + x2 cos y (dy/dx) - 2 cos x e2y (dy/dx) + sin x e2y = 6x + 2Group terms with dy/dx:(dy/dx) (x2 cos y - 2 cos x e2y) = 6x + 2 - 2x sin y - sin x e2ySolve for dy/dx:dy/dx = (6x - 2x sin y - sin x e2y + 2) / (x2 cos y - 2 cos x e2y)
Hint
Apply implicit differentiation, using the product rule for terms involving products of x and y, and the chain rule for functions of y.