Mathematicans like to call two things with the same "structure" by the same name, even if it's not obvious how they're otherwise related. In this case, the shared structure is that both the derivative of a function and D(n) follow the product rule of derivatives.
Sharing a structure means two mathematical objects are "morphic" (homomorphic[1], isomorphic[2], etc.) in some way, so in some sense they are the are same object.
In this case, we can construct a morphism. Since D(n) follows the product rule, we only need to find a function f of x for each prime p which at some x has a value of p and a derivative of 1. Then we can compose those functions by multiplication for all other natural numbers. f_p(x) = x + p is one such set of functions, giving us the complete function F_n(x) = Π_(p∈P(n)) x + p, where P(n) is the set of prime factors of n. Note, the product of the empty set is defined to be 1, so F_1(x) = 1.
Finally, the homomorphism between D and the derivatives of functions is that D(n) = F'_n(0), so in some sense, D really is a derivative.
[1]: https://en.wikipedia.org/wiki/Homomorphism
[2]: https://en.wikipedia.org/wiki/Isomorphism
In this case, we can construct a morphism. Since D(n) follows the product rule, we only need to find a function f of x for each prime p which at some x has a value of p and a derivative of 1. Then we can compose those functions by multiplication for all other natural numbers. f_p(x) = x + p is one such set of functions, giving us the complete function F_n(x) = Π_(p∈P(n)) x + p, where P(n) is the set of prime factors of n. Note, the product of the empty set is defined to be 1, so F_1(x) = 1.
Finally, the homomorphism between D and the derivatives of functions is that D(n) = F'_n(0), so in some sense, D really is a derivative.
[1]: https://en.wikipedia.org/wiki/Homomorphism
[2]: https://en.wikipedia.org/wiki/Isomorphism