How to Prove Differentiability in R2 Using the Derivative of a Function?

[raghad]

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?

 

[Svein]

I could not generalize it to prove it differentiable on R2

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

 

[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"?

 

[raghad]

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"?

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

 

[Svein]

Find the derivative and decide where it is valid.

 

[raghad]

Find the derivative and decide where it is valid.

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

 

[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.

 

[HallsofIvy]

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