Derivative

Let $f:\mathbb {Q} \rightarrow \mathbb {Q}$, and let $\epsilon :\mathbb {Q} _{>0}$. I define $\Delta _{\epsilon }f:\mathbb {Q} \rightarrow \mathbb {Q}$ by

$$(\Delta _{\epsilon }f)(x):={\frac {f(x+\epsilon )-f(x)}{\epsilon }},\quad x:\mathbb {Q} .$$

I say that $f:\mathbb {Q} \rightarrow \mathbb {Q}$ is differentiable at $x:\mathbb {Q}$, if $(\Delta _{\epsilon }f)(x)-(\Delta _{\epsilon '}f)(x)$ is nill whenever $\epsilon ,\epsilon ':\mathbb {Q} _{>0}$ are both nill.

If $f:\mathbb {Q} \rightarrow \mathbb {Q}$ is differentiable at $x:\mathbb {Q}$, then I define the derivative of $f$ at $x$ as $f'(x):=(\Delta _{\epsilon }f)(x)$, for some nill $\epsilon :\mathbb {Q} _{>0}$.