# 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 $b\neq 0$:

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

Note that

 $a^{\prime}$ $=\displaystyle\left(b\cdot\frac{a}{b}\right)^{\prime}$ $=\displaystyle b\cdot\left(\frac{a}{b}\right)^{\prime}+b^{\prime}\cdot\frac{a}% {b}$ by the Leibniz rule.

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

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

Title quotient rule for arithmetic derivative QuotientRuleForArithmeticDerivative 2013-03-22 17:04:44 2013-03-22 17:04:44 Wkbj79 (1863) Wkbj79 (1863) 4 Wkbj79 (1863) Theorem msc 11Z05