It seems from the comments that you have discovered this anyway, but the answer is yes, you get some interesting dynamics out. The derivative with all $\alpha_p=1$ goes by the name "the arithmetic derivative," and there are a few references around (Ufnarovski's being the most complete). The short version of the dynamcis story is the following: There are many numbers $n$ (e.g., primes) whose derivatives of sufficiently high order are zero. There are an easily-described set of numbers (those of the form $n=p^p$ with $p$ prime) which satisfy $n'=n$, and so $n^{(k)}=n$ for all $n$. Finally, there are many numbers (e.g., non-trivial multiples of $p^p$) whose repeated derivatives tend to $\infty$. Probably the most important question in this theory is whether or not there are any other possible orbits (i.e., non-trivial cycles).

A comment on importance: Though it's not clear to me that there's any way these links are genuinely helpful, there are some amusing ties relating statements about arithmetic derivatives to statements about other classical number theory problems. Ufnarovski shows easy links to Goldbach's conjecture and the twin primes conjecture, and some undergraduate research I supervised extended this to Sophie Germain primes and Cunningham chains.

As to the number fields case, one can certainly still play some analogous games, though it's impossible, even in nice cases, "extend" the arithmetic derivative. Even if the class group is trivial (e.g., $K=\mathbb{Q}(\sqrt{2})$), one has the obvious problem that one would like to have the prime element $\sqrt{2}$ have derivative one, but this is inconsistent with the product rule (if one wants to maintain that $2'=1$.)

I haven't thought much about other values of $\alpha_p$ (though in many cases, I'd imagine you'd get exactly the same dynamics), but as a side, let me also reference you to Buium's notion of a derivative (also going by the name arithmetic derivative) which is a little fancier (though not much more hard to define), but currently seems to be of more theoretical significance.

Yes, you get some interesting dynamics out. The derivative with all $\alpha_p=1$ goes by the name "the arithmetic derivative," and there are a few references around (Ufnarovski's being the most complete). The short version of the dynamics story is the following: There are many numbers $n$ (e.g., primes, or twice a twin prime) whose higher order derivatives $n^{(k)}$ are eventually zero. There are an easily-described set of numbers (those of the form $n=p^p$ with $p$ prime) which satisfy $n'=n$, and so $n^{(k)}=n$ for all $n$. Finally, there are many numbers (e.g., non-trivial multiples of $p^p$) with $n^{(k)}\rightarrow\infty$. A fairly major open problem is whether or not there are any other possible orbits (i.e., non-trivial cycles).

A comment on importance: Though it's not clear to me that there's any way these links are genuinely helpful, there are some amusingly sneaky ties relating statements about arithmetic derivatives to statements about other classical number theory problems. Ufnarovski gives links to Goldbach's conjecture and the twin primes conjecture, and some undergraduate research I (and colleague Ben Levitt) supervised extends this to Sophie Germain primes and Cunningham chains.

As to the number fields case, one can certainly still play some analogous games, though it's impossible, even in nice cases, to "extend" the arithmetic derivative. Even if the class group is trivial (e.g., $K=\mathbb{Q}(\sqrt{2})$), one has the obvious problem that one would like to have the prime element $\sqrt{2}$ have derivative one, but this is inconsistent with the product rule (if one wants to maintain that $2'=1$.)

I haven't thought much about other values of $\alpha_p$ (though in many cases, I'd imagine you'd get exactly the same dynamics), but as an aside, let me also reference you to Buium's notion of a derivative (also going by the name arithmetic derivative) which is a little fancier, but currently seems to be of more theoretical significance.