show that

defines a smooth function where so I am guessing it is a unit sphere.

How do we proceed with such problem?

f is a smooth function if all partial derivatives of all possible orders are defined in all points of the domain of f right?

I started with computing these partial derivatives

and these partial derivatives are all defined at any point from the domain of f

so I compute second order partial derivatives

and these 3 are also defined for all the points of the domain.

if I continue differentiating

and

which is defined for all points

What should I do next? can I conclude based on the calculations that f is certainly smooth?

Thanks in advance