example of arithmetic derivative

Consider the natural number $6$. Using the rules of the arithmetic derivative we get:
$6^{\prime }={(2\cdot 3)}^{\prime }=2^{\prime }\cdot 3+2\cdot 3^{\prime }=1\cdot 3+2\cdot 1=5$

Below is a list of the 10 first natural numbers and their first and second arithmetic derivatives:
$n$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$ $n^{\prime }$ $0$ $1$ $1$ $4$ $1$ $5$ $1$ $12$ $6$ $7$ $n^{\prime \prime }$ $0$ $0$ $0$ $4$ $0$ $1$ $0$ $16$ $5$ $1$

Title	example of arithmetic derivative
Canonical name	ExampleOfArithmeticDerivative
Date of creation	2013-03-22 13:35:12
Last modified on	2013-03-22 13:35:12
Owner	Johan (1032)
Last modified by	Johan (1032)
Numerical id	7
Author	Johan (1032)
Entry type	Example
Classification	msc 11Z05