Arithmetic derivative

The arithmetic derivative on positive integers is defined by $p\prime =1$ for every prime $p$ and the product rule $(ab)\prime =a\prime b+ab\prime$. For $n={\prod }_i{p}_i^{e_i}$, these rules give $n\prime =n{\sum }_i{e}_i/p_i$. In particular, $(p^e)\prime =e{p}^{e-1}$.