Question Detail

We need to differentiate the equation implicitly with respect to $x$.$\frac{d}{dx}(3x^2 y) + \frac{d}{dx}(xy^2) = \frac{d}{dx}(5)$Apply the product rule for differentiation to the terms involving $y$:For $3x^2 y$: $(6x)y + 3x^2 \frac{dy}{dx}$For $xy^2$: $(1)y^2 + x(2y \frac{dy}{dx})$For $5$: $0$Substitute these back into the equation:$6xy + 3x^2 \frac{dy}{dx} + y^2 + 2xy \frac{dy}{dx} = 0$Group terms containing $\frac{dy}{dx}$:$(3x^2 + 2xy) \frac{dy}{dx} = -6xy - y^2$Isolate $\frac{dy}{dx}$:$\frac{dy}{dx} = \frac{-6xy - y^2}{3x^2 + 2xy}$Factor out $-1$ from the numerator:$\frac{dy}{dx} = -\frac{y^2 + 6xy}{3x^2 + 2xy}$This matches option B.