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arithmetic derivative
The arithmetic derivative $n^{{\prime}}$ of a natural number $n$ is defined by the following rules:

$p^{{\prime}}=1$ for any prime $p$.

$(ab)^{{\prime}}=a^{{\prime}}b+ab^{{\prime}}$ for any $a,b\in\mathbb{N}$ (Leibniz rule).
To define the arithmetic derivative of a negative number, we first note that $1^{{\prime}}=0$ by the Leibniz rule ($1^{{\prime}}=(1\cdot 1)^{{\prime}}=1\cdot 1^{{\prime}}+1^{{\prime}}\cdot 1=2% \cdot 1^{{\prime}}$, so $1^{{\prime}}=0$), and further that we must have
$\displaystyle 0=1^{{\prime}}=\left((1)\cdot(1)\right)^{{\prime}}=2\cdot(1)% ^{{\prime}},$ 
so $(1)^{{\prime}}=0$. The product rule now requires that we define $(n)^{{\prime}}=(n^{{\prime}})+n(1)^{{\prime}}=(n^{{\prime}})$.
Further, we can extend this definition to rational numbers by insisting that the quotient rule holds, i.e. for a prime $p$ we should have
$\displaystyle 0=1^{{\prime}}=\left(p\cdot\frac{1}{p}\right)^{{\prime}}=\left(% \frac{1}{p}\right)^{{\prime}}p+\frac{1}{p},$ 
giving us that
$\left(\frac{1}{p}\right)^{{\prime}}=\frac{1}{p^{2}},$ 
i.e. the usual quotient rule from calculus. We now complete the definition by extending multiplicatively (i.e. using the Leibniz rule).
The arithmetic derivatives for the first few positive integers are 0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, 8, 32, 1, 21, 1, 24, 10, 13, 1, 44, 10, etc.
As a consequence of $p^{{\prime}}=1$ for a prime $p$, the arithmetic derivative of a semiprime (whether squarefree or not) works out to $(pq)^{{\prime}}=p^{{\prime}}q+pq^{{\prime}}=1p+q1=p+q$. For example, the arithmetic derivative of 10 is 7, which is 2 plus 5.
The only cases of $n^{{\prime}}=n$ for $1<n<1024$ are 0, 4, 27.
$n$  $n^{{\prime}}$  $n$  $n^{{\prime}}$  $n$  $n^{{\prime}}$  $n$  $n^{{\prime}}$  $n$  $n^{{\prime}}$  $n$  $n^{{\prime}}$  $n$  $n^{{\prime}}$  $n$  $n^{{\prime}}$  $n$  $n^{{\prime}}$  $n$  $n^{{\prime}}$ 
0  0  10  7  20  24  30  31  40  68  50  45  60  92  70  59  80  176  90  123 
1  0  11  1  21  10  31  1  41  1  51  20  61  1  71  1  81  108  91  20 
2  1  12  16  22  13  32  80  42  41  52  56  62  33  72  156  82  43  92  96 
3  1  13  1  23  1  33  14  43  1  53  1  63  51  73  1  83  1  93  34 
4  4  14  9  24  44  34  19  44  48  54  81  64  192  74  39  84  124  94  49 
5  1  15  8  25  10  35  12  45  39  55  16  65  18  75  55  85  22  95  24 
6  5  16  32  26  15  36  60  46  25  56  92  66  61  76  80  86  45  96  272 
7  1  17  1  27  27  37  1  47  1  57  22  67  1  77  18  87  32  97  1 
8  12  18  21  28  32  38  21  48  112  58  31  68  72  78  71  88  140  98  77 
9  6  19  1  29  1  39  16  49  14  59  1  69  26  79  1  89  1  99  75 
References
 1 EJ Barbeau, “Remark on an arithmetic derivative”. Can. Math. Bull. 4 (1961): 117  122
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Comments
What is it good for?
While browsing the encyclopaedia, I came across the definition of derivative of a number. It is certainly an inteteresting illustration of how Leibniz's rule can be used to define a notion of derivative in a context where one would ordinarily not think of differential calculus. My question to you is "What is this notion of integer derivative good for?", meaning "Are you aware of any theorems of number theory which can be proven using the integer derivative?".
Re: What is it good for?
... when the derivative is zero, the number is a
constant :)
Matte
Re: What is it good for?
What exactly do you mean by this??? I would like to think that every single integer is a constant.
Johan
Re: What is it good for?
Of course every integer is a constant.
My comment was just a joke.
Nevertheless, I'm too curious how
this integer derivative is used.
Matte
Re: What is it good for?
Apparently Goldbach's conjecture can be stated using
the number derivative as:
if a\in Z then there exists a b\in Z such that
b'=2a.
This is mentioned in
http://web.mit.edu/lwest/www/intmain.pdf
http://www.maa.org/mathland/mathtrek_03_22_04.html
The number derivative is also related to the
twin prime conjecture:
http://www.lacim.uqam.ca/~plouffe/JIS/ufnarovski.pdf
Matte
Re: What is it good for?
It's unfortunate that this number derivative is not compatible with addition. In general (n+m)' != n' + m'. If it were, it would have made a nice example of a differential ring (field) if extended to the integers (rationals).