quotient rule for arithmetic derivative

Theorem.

If the notion of arithmetic derivative is extended to rational numbers, then we have that, for every $a,b\\in\\mathbb{Z}$ with $b\eq 0$ :

\\left(\\frac{a}{b}\\right)^{\\prime}=\\frac{a^{\\prime}b-b^{\\prime}a}{b^{2}}

Proof.

Note that

Thus,

$\\begin{array}[]{rl}\\displaystyle b\\cdot\\left(\\frac{a}{b}\\right)^{\\prime}&=%\\displaystyle a^{\\prime}-b^{\\prime}\\cdot\\frac{a}{b}\\\\&\\\\&=\\displaystyle\\frac{a^{\\prime}b-b^{\\prime}a}{b}.\\end{array}$

It follows that

\\left(\\frac{a}{b}\\right)^{\\prime}=\\frac{a^{\\prime}b-b^{\\prime}a}{b^{2}}.

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