I have stumped upon a question about whether one can generalize the notion of derivatives onto a function such as $x mapsto x^{frac 13}$ at $x=0$.

Spicifically, can we compare infinite derivatives to each other?

I am currently working on the theory of infinite divergent integrals/series (divergent quantities), and in that theory I introduce the notion of “germs”, which is a generalization of limits. In finite case they give usual limits, but at a pole of a function they are infinite quantities, containing information of growth rate and the regularized value. They are defined via divergent integrals.

I only had considered germs of monomials and inverses of monomials of integer powers though.

Still, if to generalize the formula to a fractional case, we formally obtain the derivative of $f(x)=x^{1/3}$ at $x=0$ as

$$f'(0)=frac{omega _+^{5/3}-omega _-^{5/3}}{3 Gamma left(frac{8}{3}right)}$$

where $omega_+=sum_{0}^infty 1=int_{-1/2}^infty dx$ and $omega_-=omega_+-1=sum_1^infty 1=int_{1/2}^infty dx$ are two divergent integrals/series.

This is an infinite quantity, but comparable to other infinite quatities as well. Its regularized value is zero.

Thus, I wonder, whether it would make sense to speak about infinite derivatives?