Question

If the function \( u = \sin^{-1} \left( \frac{x^3 + y^3}{x + y} \right) \), then \[ x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} \] is

• A. \( 2 \tan u \)
• B. \( 3 \csc u \)
• C. \( 3 \sin u \)
• D. \( 2 \cos u \)

Hint:

Use Euler's theorem for homogeneous functions: sum of variables times their partial derivatives equals the degree times the function.

Answer 1

The Correct Option is A

By Euler's theorem for homogeneous functions: If \( u \) is a function of degree \( n \), \[ x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = n u \] Now, \[ f(x, y) = \frac{x^3 + y^3}{x + y} \] Both numerator and denominator are homogeneous functions: Numerator degree = 3, Denominator degree = 1 So, total degree = 3 - 1 = 2 Thus, \[ x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = 2 \times \frac{du}{du} = 2 \times \frac{d}{du} \left( \sin^{-1} \left( \frac{x^3 + y^3}{x + y} \right) \right) \] Simplifying, this results in: \[ = 2 \tan u \]