Homework Help: Differentiability in R2

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1. Feb 18, 2015

Let U={(x,y) in R2:x2+y2<4}, and let f(x,y)=√.(4−x2−y2)
Prove that f is differentiable, and find its derivative.

I do know how to prove it is differentiable at a specific point in R2, but I could not generalize it to prove it differentiable on R2. Any hint?

2. Feb 18, 2015

Svein

As far as I can see you are not asked to do that, you are asked to prove differentiability on U.

3. Feb 18, 2015

HallsofIvy

As Svein said, you are not asked to prove differentiability on R2, you are asked to prove it on the given set, U. Now, what is the definition of "differentiable on a set A"?

4. Feb 19, 2015

I know the definition of "differentiable at a point" , but i am not sure of the definition of differentiability on a set. Does it have to do with end points? i am stuck in this question and your help is much appreciated

5. Feb 19, 2015

Svein

Find the derivative and decide where it is valid.

6. Feb 19, 2015

Can i pick an arbitrary subset of U and prove that the function is differentiable there then conclude that it is differentiable on U ?

7. Feb 19, 2015

LCKurtz

Why not just pick any (x,y) in the domain U and see if it works there? U is an open domain so there are no boundary points U to worry about.

8. Feb 20, 2015

HallsofIvy

A function is differentiable "on a set U" if and only if it is differentiable at every point in U!