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The arithmetic derivative on positive integers is specified by $p^{\prime }=1$ for each prime number $p$ and the product rule $(ab{)}^{\prime }=a^{\prime }b+a{b}^{\prime }$. These conditions force $1^{\prime }=0$. For a prime factorization $n={\prod }_i{p}_i^{e_i}$ they giveRepeated use of the product rule proves necessity. Conversely, the fundamental theorem of arithmetic makes this formula well-defined, and the addition of prime exponents under multiplication verifies the product rule. Thus it proves existence as well as uniqueness.^[ref]

For example, $12^{\prime }=12(2/2+1/3)=16$. Despite its name, the arithmetic derivative does not preserve addition: $(2+3{)}^{\prime }=5^{\prime }=1$, whereas $2^{\prime }+3^{\prime }=2$. Its analogy with an ordinary derivative preserves the product rule while sacrificing the usual sum rule.

Bo Åhlander coauthored a study of the arithmetic derivative with Victor Ufnarovski. The operation is determined on positive integers by assigning one to each prime number and imposing the product rule. Thus the derivative of $6=2\cdot 3$ is $3+2=5$, illustrating an arithmetic analogue of differentiation built from integer factorization.^[ref]

Victor Ufnarovski studied the arithmetic derivative with Bo Åhlander. Their definition assigns derivative one to every prime number and extends by the product rule. For $n=\prod p_i^{a_i}$, the resulting formula $n{\sum }_i{a}_i/p_i$ turns differentiation into an operation governed by integer factorization, rather than an ordinary derivative of a real-variable function.^[ref]