7. (10 Points) Let $f:\mathbb{R}^2\to\mathbb{R}$ be a function. Assume that all second partial derivatives of $f$ exist and continuous and

$$f_{xx}+f_{yy}=0,\quad \text{for all }(x,y)\in\mathbb{R}^2.$$

Define $g:\mathbb{R}^2\to\mathbb{R}$ by $g(u,v)=f(u^2-v^2,2uv)$. Find $g_{uu}+g_{vv}$.