Question Detail

This problem requires implicit differentiation. Differentiate both sides of the equation with respect to x, treating y as a function of x (y(x)) and applying the product rule and chain rule where necessary.1. d⁄dx(3x2y): Using the product rule, this becomes (d⁄dx(3x2)) * y + 3x2 * (d⁄dx(y)) = 6xy + 3x2 (dy⁄dx).2. d⁄dx(xy2): Using the product rule, this becomes (d⁄dx(x)) * y2 + x * (d⁄dx(y2)) = 1 * y2 + x * (2y * dy⁄dx) = y2 + 2xy (dy⁄dx).3. d⁄dx(5): The derivative of a constant is 0.Combine these results: 6xy + 3x2 (dy⁄dx) + y2 + 2xy (dy⁄dx) = 0. Now, group all terms containing dy⁄dx on one side and move the other terms to the other side: (dy⁄dx) (3x2 + 2xy) = -6xy - y2. Factor out a negative sign on the right side: (dy⁄dx) (3x2 + 2xy) = -(6xy + y2). Finally, solve for dy⁄dx: dy⁄dx = -(y2 + 6xy)⁄(3x2 + 2xy). This matches option B.