Question Detail

This requires implicit differentiation. Differentiate both sides of the equation with respect to x:d/dx(3x2y) + d/dx(xy2) = d/dx(5)For the first term, 3x2y, use the product rule: d/dx(uv) = u'v + uv'Let u = 3x2, u' = 6xLet v = y, v' = dy/dxSo, d/dx(3x2y) = 6xy + 3x2(dy/dx)For the second term, xy2, use the product rule again:Let u = x, u' = 1Let v = y2, v' = 2y(dy/dx)So, d/dx(xy2) = 1y2 + x2y(dy/dx) = y2 + 2xy(dy/dx)For the right side, d/dx(5) = 0.Combine all differentiated terms:6xy + 3x2(dy/dx) + y2 + 2xy(dy/dx) = 0Group terms with dy/dx on one side and other terms on the other side:dy/dx (3x2 + 2xy) = -6xy - y2Factor out a negative sign from the right side:dy/dx (3x2 + 2xy) = -(6xy + y2)Finally, solve for dy/dx:dy/dx = -(y2 + 6xy) / (3x2 + 2xy)This matches option B.