Question Detail

We need to use implicit differentiation with respect to x.Differentiate each term:1. d/dx (3x2y): Using the product rule: (d/dx [3x2])y + 3x2(d/dx [y]) = 6xy + 3x2(dy/dx)2. d/dx (xy2): Using the product rule: (d/dx [x])y2 + x(d/dx [y2]) = 1 * y2 + x * (2y * dy/dx) = y2 + 2xy(dy/dx)3. d/dx (5) = 0Combine the differentiated terms:6xy + 3x2(dy/dx) + y2 + 2xy(dy/dx) = 0Group terms with dy/dx on one side and other terms on the other side:(3x2 + 2xy)(dy/dx) = -6xy - y2Solve for dy/dx:dy/dx = (-6xy - y2) / (3x2 + 2xy)Factor out a negative sign from the numerator:dy/dx = -(6xy + y2) / (3x2 + 2xy)This matches option B.