Question Detail

Differentiate implicitly with respect to ( x ):( \frac{d}{dx}(3x^2 y) + \frac{d}{dx}(xy^2) = \frac{d}{dx}(5) )Using the product rule for each term on the left:( (6xy + 3x^2 \frac{dy}{dx}) + (y^2 + x \cdot 2y \frac{dy}{dx}) = 0 )( 6xy + 3x^2 \frac{dy}{dx} + y^2 + 2xy \frac{dy}{dx} = 0 )Group terms containing ( \frac{dy}{dx} ):( (3x^2 + 2xy) \frac{dy}{dx} = -6xy - y^2 )Solve for ( \frac{dy}{dx} ):( \frac{dy}{dx} = \frac{-6xy - y^2}{3x^2 + 2xy} = -\frac{y^2 + 6xy}{3x^2 + 2xy} )This matches option B.