The Chain Rule & Implicit Differentiation · Challenge problem

For $x^2 > 4y$, let $u(x,y)$ be the larger root of the quadratic $t^2 - xt + y = 0$, so $u$ is a smooth function of the coefficients. Using implicit differentiation, find $\dfrac{\partial^2 u}{\partial x\,\partial y}$ at $(x, y) = (5, 6)$.