If u=tan-1 ((x2+y2)/(x+y)), then x ∂u/∂x+y ∂u/∂y=

To solve this problem, we need to use the chain rule of differentiation.

First, let's find the partial derivatives of u with respect to x and y.

∂u/∂x = (1/(1+((x^2+y^2)/(x+y))^2)) * (2x(x+y) - (x^2+y^2))/(x+y)^2

∂u/∂y = (1/(1+((x^2+y^2)/(x+y))^2)) * (2y(x+y) - (x^2+y^2))/(x+y)^2

Now, we can substitute these into the equation x ∂u/∂x + y ∂u/∂y.

x ∂u/∂x + y ∂u/∂y = x * [(1/(1+((x^2+y^2)/(x+y))^2)) * (2x(x+y) - (x^2+y^2))/(x+y)^2] + y * [(1/(1+((x^2+y^2)/(x+y))^2)) * (2y(x+y) - (x^2+y^2))/(x+y)^2]

This is the final answer.