If I want to take the derivative of $ax^n$, I will get $anx^{n-1}$. If I were to take the derivative again, I get $an(n-1)x^{n-2}$.
We can generalize this for integer k easily to get the kth derivative $a\frac{n!}{(n-k)!} x ^{n-k}$. But what about for a more general k?
Does this have some name? Has it been widely studied? If so, can you show how to generalize this formula for kth derivative of $ax^n$, and explain how it works? If not, is there a good reason it is impossible?
To expand on Jonas's comment: Yes, it makes sense. For the case of the power function, one can consider
$$\frac{\Gamma(n+1)}{\Gamma(n-\alpha+1)}x^{n-\alpha}$$
as the $\alpha$-th derivative of the power function $x^n$, where $\Gamma(z)$ is the gamma function, the generalization of the factorial to the complex plane.
In general, one has a number of definitions for so-called "fractional derivatives", or, as Spanier and Oldham prefer to call it, the "differintegral". Negative values of $\alpha$ in expressions like the one given above correspond to integration, positive values correspond to differentiation, and in general $\alpha$ can be complex.
There's a lot of things to look at (Caputo derivatives, Riemann-Liouville integrals, Grunwald-Lednikov series), and I suggest you look at the book I linked to first, and then search around the web. Have fun!