Differentiability in $R^2$ 
Let $U=\{(x,y) \text{ in } \mathbb{R}^2 : x_2 + y_2 < 4\}$, and let $f(x,y)= \sqrt{4-x_2-y_2}$.
  Prove that $f$ is differentiable, and find its derivative.

I do know how to prove it is differentiable at a specific point in $\mathbb{R}^2$, but I could not generalize it to prove it differentiable on $\mathbb{R}^2$. Any hint? 
 A: A function from $\mathbb{R^{2}}\to\mathbb{R}$ is said to be differentiable at a point $(a,b)$ iff there exist $\phi(h,k)$ and $\psi(h,k)$ from $\mathbb{R}^{2}\to\mathbb{R}$ such that
$f(a+h,b+k)=h\frac{\partial f}{\partial x}\vert_{(a,b)}+k\frac{\partial f}{\partial y}\vert_{(a,b)}+h\phi(h,k)+k\psi(h,k)$ such that  $\phi(h,k)\to 0\quad \text{and}\quad\psi(h,k)\to 0$ as $(h,k)\to(0,0)$.
The existence and continuity of the partial derivatives give only a sufficient condition for differentiability.
For example $f(x,y)=x^{2}\sin(\frac{1}{x})+y^{2}\sin(\frac{1}{y})$ is differentiable . But the partial derivatives are not continuous. To see that it is differentiable it is easy to note that the partial derivatives at $(0,0)$ is $0$. And take $\phi(h,k)=h\sin(\frac{1}{h})$ and $\psi(h,k)=k\sin(\frac{1}{k})$.
A: A function is differentiable in $U$ if it is differentiable at every point in $U$. So what you need to do is pick an arbitrary $(a,b) \in U$ and show that $f$ is differentiable at $(a,b)$.
