The problem is stated as follows:
The function is defined in as and if .
In what points is differentiable?
It's easily shown that the two first partial derivatives are continuous except at the origin, so f is differentiable at every point that isn't the origin.
Using the definition of differentiability for the origin,
After trying Maclaurin expansions of the numerator and denominator without getting anywhere (perhaps due to arithmetical mistakes), and fooling around a bit without any more luck, I switched to polar coordinates, i.e. . Since r approaches 0 regardless of the angle , the limit is equivalent to showing that .
By using L'H˘pital's rule three times, the latter limit can be shown, and while the operations are simple, a considerable amount of arithmetic is required to show differentiability in the origin (and hence, in every point in ) like this. I strongly suspect that I have missed at least one much easier solution to this. Hints?