Question Detail

Differentiate each term with respect to x, remembering to apply the product rule and chain rule where necessary.d/dx (x² sin y) - d/dx (e2y cos x) = d/dx (3x² + 2x - 5)1. For d/dx (x² sin y): 2x sin y + x² cos y (dy/dx) (Product Rule)2. For d/dx (e2y cos x): (d/dx e2y) cos x + e2y (d/dx cos x) (Product Rule) = (2e2y dy/dx) cos x + e2y (-sin x) = 2e2y cos x (dy/dx) - e2y sin x3. For d/dx (3x² + 2x - 5): 6x + 2Combine these results:2x sin y + x² cos y (dy/dx) - [2e2y cos x (dy/dx) - e2y sin x] = 6x + 22x sin y + x² cos y (dy/dx) - 2e2y cos x (dy/dx) + e2y sin x = 6x + 2Group terms with dy/dx on one side and other terms on the other side:(x² cos y - 2e2y cos x) dy/dx = 6x + 2 - 2x sin y - e2y sin xSolve for dy/dx:dy/dx = (6x + 2 - 2x sin y - e2y sin x) / (x² cos y - 2e2y cos x)This matches option C.