The Chain Rule & Implicit Differentiation · Superchallenge problem

The map $u = x + y^3$, $v = y + x^3$ sends $(x,y) = (1,1)$ to $(u,v) = (2,2)$ and is invertible near that point. Regard $x$ and $y$ as functions of $(u,v)$ there. Find $\dfrac{\partial^2 x}{\partial u^2}$ at $(u,v) = (2,2)$.