quotient rule for arithmetic derivative

Theorem.

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

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

Proof.

Note that

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

Thus,

It follows that

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

∎

Title	quotient rule for arithmetic derivative
Canonical name	QuotientRuleForArithmeticDerivative
Date of creation	2013-03-22 17:04:44
Last modified on	2013-03-22 17:04:44
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	4
Author	Wkbj79 (1863)
Entry type	Theorem
Classification	msc 11Z05