The Chain Rule & Implicit Differentiation · Challenge problem

Let $z = x^2 + y^2$ and $w = xy$. Regard $z$ as a function of $x$ and $w$ (eliminate $y$). Find $\left(\dfrac{\partial z}{\partial x}\right)_{w}$, the rate of change of $z$ with respect to $x$ with $w$ held constant, at the point $x = 1$, $y = 2$.