Question Detail

Differentiate each term of x2 sin y - e2y cos x = 3x2 + 2x - 5 with respect to x using the product rule and chain rule where necessary.1. d/dx(x2 sin y): (2x)sin y + x2(cos y dy/dx) = 2x sin y + x2 cos y dy/dx2. d/dx(-e2y cos x): -[(d/dx(e2y))cos x + e2y(d/dx(cos x))] = -[(2e2y dy/dx)cos x + e2y(-sin x)] = -2e2y cos x dy/dx + e2y sin x3. d/dx(3x2 + 2x - 5): 6x + 2Combine the derivatives:2x sin y + x2 cos y dy/dx - 2e2y cos x dy/dx + e2y sin x = 6x + 2Group terms with dy/dx:(x2 cos y - 2e2y cos x) dy/dx = 6x + 2 - 2x sin y - e2y sin xSolve for dy/dx:dy/dx = (6x - 2x sin y - e2y sin x + 2) / (x2 cos y - 2e2y cos x).Note that sin x e2y is the same as e2y sin x.