Question Detail

To find dy/dx for the implicit equation 3x²y + xy² = 5, we differentiate both sides with respect to x.1. Differentiate 3x²y using the product rule: d/dx(3x²y) = (d/dx(3x²))y + 3x²(d/dx(y)) = 6xy + 3x² (dy/dx)2. Differentiate xy² using the product rule and chain rule: d/dx(xy²) = (d/dx(x))y² + x(d/dx(y²)) = 1*y² + x(2y dy/dx) = y² + 2xy (dy/dx)3. Differentiate the constant 5: d/dx(5) = 0Combine the derivatives:6xy + 3x² (dy/dx) + y² + 2xy (dy/dx) = 0Group terms containing dy/dx on one side and other terms on the other side:(3x² + 2xy) (dy/dx) = -6xy - y²Factor out a negative sign from the right side:(3x² + 2xy) (dy/dx) = -(6xy + y²)Solve for dy/dx:dy/dx = -(y² + 6xy) / (3x² + 2xy)Thus, option B is correct.