Question Detail

This requires implicit differentiation. Differentiate each term of the equation with respect to x, remembering to apply the chain rule for terms involving y (i.e., dy⁄dx will appear).1. d⁄dx(x2 sin y): Using the product rule, this becomes 2x sin y + x2 cos y (dy⁄dx).2. d⁄dx(-e2y cos x): Using the product rule, this becomes -[ (d⁄dx(e2y)) cos x + e2y (d⁄dx(cos x)) ] = -[ (e2y * 2 * dy⁄dx) cos x + e2y (-sin x) ] = -2e2y cos x (dy⁄dx) + e2y sin x.3. d⁄dx(3x2 + 2x - 5): This becomes 6x + 2.Combine these: 2x sin y + x2 cos y (dy⁄dx) - 2e2y cos x (dy⁄dx) + e2y sin x = 6x + 2. Now, group all terms containing dy⁄dx on one side and move the other terms to the other side: (dy⁄dx) (x2 cos y - 2e2y cos x) = 6x + 2 - 2x sin y - e2y sin x. Finally, solve for dy⁄dx: dy⁄dx = (6x + 2 - 2x sin y - e2y sin x) ⁄ (x2 cos y - 2e2y cos x). This matches option C.