Question

If \( (x + y) \sin u = x^2 y^2 \), then \( x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} \) is

• A. \( \sin u \)
• B. \( \cos u \)
• C. \( 2 \tan u \)
• D. \( \tan u \)

Hint:

Use implicit differentiation to find the derivatives of a function involving both \( x \) and \( y \).

Answer 1

The Correct Option is D

Step 1: Differentiate the given equation.
Differentiate \( (x + y) \sin u = x^2 y^2 \) with respect to \( x \) and \( y \) using implicit differentiation. Then solve for \( x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} \).

Step 2: Conclusion.
Thus, the correct answer is option (D).

Final Answer: \[ \boxed{\text{(D) } \tan u} \]