Number Derivative: A General Pursuit

[dodo]

There is something I don't understand about the recent concept of a "number derivative". It seems to me that the (very illusorious) 'relation' to the real derivative is driving the interest, while a more general pursuit should be in place.

I mean, any function satisfying the Leibniz rule, f(ab) = a f(b) + b f(a) will comply with the following,
$$\begin{array}{ll} \bullet & f(a^n) = n a^{n-1} f(a) \\ \bullet & f(n) = n \sum_{i=1}^k e_i \frac{f(p_i)}{p_i} \,, \quad \mbox{where } n = p_1^{e_1} p_2^{e_2} ... p_k^{e_k} \end{array}$$
without any need of defining f(p) = 1 for p prime, which is just a particular case.

 

[fresh_42]

Anytime when "derivative" is used, it is basically the Leibniz rule. It is the essence of considering derivatives, since it gives instructions how multiplication turns into addition, and linearization is why we consider derivatives in the first place.

Now your specific example is defined to investigate a specific area, which means that additional conditions might be useful for certain purposes. Otherwise there might be too many derivatives to be of use. Will say: this is an arbitrary condition for this specific case.

E.g. Lie multiplication is a "derivative", too. It obeys the Leibniz rule, which is called Jacobi identity in this case. However, we also demand $[X,X]=0$.