Question

If \(u=\dfrac{xy}{x+y}\), then \(x\dfrac{\partial u}{\partial x}+y\dfrac{\partial u}{\partial y}\) is equal to

• A. \(1\)
• B. \(u\)
• C. \(-u\)
• D. \(0\)

Hint:

Remember Euler's theorem for homogeneous functions: \[ \boxed{ x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} = nu } \] where \(n\) is the degree of homogeneity. If the degree is \(1\), the result is simply \[ \boxed{ x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} = u. } \]

Answer 1

The Correct Option is B