Answer is: option2
\( \frac{5 - 2xy - 2y^2}{x^2 + 4xy} \)Solution:
1. Differentiate \( x^2 y \) using the product rule: \[ \frac{d}{dx} (x^2 y) = x^2 \frac{dy}{dx} + 2xy \] 2. Differentiate \( 2xy^2 \) using the product rule: \[ \frac{d}{dx} (2xy^2) = 2x \cdot 2y \frac{dy}{dx} + 2y^2 \] \[ = 4xy \frac{dy}{dx} + 2y^2 \] 3. Differentiate the right-hand side: \[ \frac{d}{dx} (5x) = 5 \] Now, combine all terms: \[ (x^2 \frac{dy}{dx} + 2xy) + (4xy \frac{dy}{dx} + 2y^2) = 5 \] \[ x^2 \frac{dy}{dx} + 4xy \frac{dy}{dx} + 2xy + 2y^2 = 5 \] \[ (x^2 + 4xy) \frac{dy}{dx} = 5 - 2xy - 2y^2 \] Solving for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{5 - 2xy - 2y^2}{x^2 + 4xy} \] Final Answer: \[ \boxed{\frac{5 - 2xy - 2y^2}{x^2 + 4xy}} \]