Question Detail

To find dy/dx for the implicit equation 3x2y + xy2 = 5, we differentiate each term with respect to x, remembering to use the product rule and chain rule for terms involving y.1. Differentiate 3x2y: d/dx(3x2y) = (d/dx(3x2))y + 3x2(d/dx(y)) = 6xy + 3x2(dy/dx)2. Differentiate xy2: d/dx(xy2) = (d/dx(x))y2 + x(d/dx(y2)) = 1*y2 + x(2y dy/dx) = y2 + 2xy(dy/dx)3. Differentiate 5: d/dx(5) = 0Combine the derivatives:(6xy + 3x2(dy/dx)) + (y2 + 2xy(dy/dx)) = 0Group terms containing dy/dx:3x2(dy/dx) + 2xy(dy/dx) = -6xy - y2Factor out dy/dx:(dy/dx)(3x2 + 2xy) = -(6xy + y2)Finally, solve for dy/dx:dy/dx = -(y2 + 6xy) / (3x2 + 2xy)This matches option B.