Answer
As a result $w_u=\frac{-e^v}{(u+v^2)^2}$, $w_v=\frac{e^v(u+v^2)-2ve^v}{(u+v^2)^2}$.
Work Step by Step
$w=\frac{e^v}{u+v^2}$
In order to find $w_u$ we treat $v$ as a constant and differentiate with respect to $u$.
$w_u=\frac{-e^v}{(u+v^2)^2}$
In order to find $w_v$ we treat $u$ as a constant and differentiate with respect to $v$.
$w_v=\frac{e^v(u+v^2)-2ve^v}{(u+v^2)^2}$
