If u = sin⁻¹((x + 2y) / (x⁸ + y⁸)), then what is the value of x(∂u/∂x) + y(∂u/∂y)?
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Correct answer: (D) −7 tan(u)
Explanation
u = sin⁻¹(f), where f = (x + 2y)/(x⁸ + y⁸) is homogeneous of degree 1 − 8 = −7. By the corollary of Euler's theorem, x(∂u/∂x) + y(∂u/∂y) = n·f(u)/f'(u) = −7 sin u/cos u = −7 tan u.