It looks like you're supposed to show
[tex]u_x+u_y=u[/tex]
We have
[tex]u=\dfrac{e^{x+y}}{e^x+e^y}[/tex]
In the numerator, we have [tex]e^{x+y}=e^xe^y[/tex], so that the derivative wrt either [tex]x[/tex] or [tex]y[/tex] is simply [tex]e^xe^y=e^{x+y}[/tex]. In the denominator, either [tex]e^x[/tex] or [tex]e^y[/tex] vanishes.
Differentiating wrt [tex]x[/tex] gives, by the quotient rule,
[tex]u_x=\dfrac{e^{x+y}(e^x+e^y)-e^{x+y}e^x}{(e^x+e^y)^2}=\dfrac{e^{x+y}e^y}{(e^x+e^y)^2}[/tex]
Similarly, differentiating wrt [tex]y[/tex] gives
[tex]u_y=\dfrac{e^{x+y}e^x}{(e^x+e^y)^2}[/tex]
Then
[tex]u_x+u_y=\dfrac{e^{x+y}(e^y+e^x)}(e^x+e^y)^2}[/tex]
[tex]u_x+u_y=\dfrac{e^{x+y}}e^x+e^y}=u[/tex]
as required.
