Question

If \( u = \tan^{-1} \left( \frac{x^3 + y^2}{x + y} \right) \), then \( \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} \) is:

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

Hint:

When differentiating inverse trigonometric functions, use the chain rule and simplify the result carefully.

Answer 1

The Correct Option is A

Step 1: Compute partial derivatives. We differentiate the given function \( u = \tan^{-1} \left( \frac{x^3 + y^2}{x + y} \right) \) with respect to both \( x \) and \( y \).
Step 2: Conclusion. After performing the differentiation, we obtain \( \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = \sin 2u \).