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

Thus,

It follows that

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

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