Question Detail

To differentiate implicitly, differentiate both sides of the equation 3x²y + xy² = 2y with respect to x.Using the product rule for 3x²y: 6xy + 3x²(dy/dx)Using the product rule for xy²: y² + x(2y)(dy/dx)Differentiating 2y: 2(dy/dx)Combining: 6xy + 3x²(dy/dx) + y² + 2xy(dy/dx) = 2(dy/dx)Group terms with dy/dx: 3x²(dy/dx) + 2xy(dy/dx) - 2(dy/dx) = -6xy - y²Factor out dy/dx: dy/dx (3x² + 2xy - 2) = -(6xy + y²)Solve for dy/dx: dy/dx = -(6xy + y²) / (3x² + 2xy - 2)Multiplying numerator and denominator by -1: dy/dx = (6xy + y²) / (2 - 3x² - 2xy)This matches option D.