For questions concerning the arithmetic derivative from number theory.
The arithmetic derivative of an integer is defined recursively. Let $n’$ denote the arithmetic derivative of $n$. Then, for prime $p$, and integers $a$, $b$, the following hold:
$$p’=1,$$ $$(ab)’=(a’)b+a(b’).$$
Note that the second rule is analogous to the product rule for usual derivatives.
Arithmetic derivatives can also be extended to the rationals, by adding a third rule, analogous to the usual quotient rule:
$$\left(\frac{a}{b}\right)’=\frac{(a’)b-a(b’)}{b^2}.$$
Reference: Arithmetic Derivative.