Question Detail

We use implicit differentiation. Differentiate both sides of the equation 3x2y + xy2 = 5 with respect to x.For d/dx (3x2y), apply the product rule: 3 ∙ (2x ∙ y + x2 ∙ dy/dx) = 6xy + 3x2 dy/dx.For d/dx (xy2), apply the product rule: 1 ∙ y2 + x ∙ (2y ∙ dy/dx) = y2 + 2xy dy/dx.The derivative of the constant 5 is 0.So, 6xy + 3x2 dy/dx + y2 + 2xy dy/dx = 0.Group terms with dy/dx:dy/dx (3x2 + 2xy) = -6xy - y2dy/dx (3x2 + 2xy) = -(6xy + y2).Finally, solve for dy/dx:dy/dx = -(y2 + 6xy) / (3x2 + 2xy). This matches option B.