Well, I can't speak to what's wrong with you! But here, you have a sign error.
$\displaystyle \nabla\times F= \left|\begin{array}{ccc}\vec{ix} & \vec{j} & \vec{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ \frac{-y}{x^2+y^2} & \frac{x}{x^2+y^2} & 0 \end{array}\right|$
$\displaystyle = \left(\frac{\partial}{\partial x} \left(\frac{x}{x^2+ y^2}\right)- \frac{\partial}{\partial y}\frac{-y}{x^2+y^2}\right)\vec{k}$
With those two "-" signs, the partial derivatives add:
$\displaystyle \left(\frac{y^2- x^2}{(x^2+ y^2)}+ \frac{y^2- x^2}{(x^2+ y^2)}\right)\vec{k}= 2\frac{y^2- x^2}{(x^2+ y^2)}\vec{k}$