Question Detail

To find dy/dx for the implicit equation 3x2y + xy2 = 5, we differentiate both sides with respect to x.Apply the product rule for differentiation for each term on the left side: (uv)' = u'v + uv'.For 3x2y: d/dx(3x2y) = (d/dx(3x2))y + 3x2(d/dx(y)) = 6xy + 3x2(dy/dx).For xy2: d/dx(xy2) = (d/dx(x))y2 + x(d/dx(y2)) = 1*y2 + x(2y(dy/dx)) = y2 + 2xy(dy/dx).For 5: d/dx(5) = 0.Combine these derivatives:(6xy + 3x2 dy/dx) + (y2 + 2xy dy/dx) = 0Group terms containing dy/dx:3x2 dy/dx + 2xy dy/dx = -6xy - y2Factor out dy/dx:(3x2 + 2xy) dy/dx = -(y2 + 6xy)Solve for dy/dx:dy/dx = -(y2 + 6xy) / (3x2 + 2xy).