Derivative

Let $f:\mathbb {Q} \rightarrow \mathbb {Q}$ be a function, 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 $\epsilon :\mathbb {Q} _{>0}$.

Criticism

This definition makes things way more complicated. I will demonstrate this with the following example. Let's suppose that $f(x)=x^{3}$ so $(\Delta _{\epsilon }f)(x)=3x^{2}+3\epsilon x+\epsilon ^{2}$ and $(\Delta _{\epsilon }f-\Delta _{\epsilon '}f)(x)=3(\epsilon -\epsilon ')x+(\epsilon ^{2}-\epsilon '^{2})$. And let's say that we are in a context where anything with absolute value below 0.1 is nill. Let $\epsilon =0.091$ and $\epsilon '=0.001$. Then $(\Delta _{\epsilon }f-\Delta _{\epsilon '}f)(x)=0.27x+0.00828$. This quantity is greater than 0.1, and thus non-nill, when $x>0.339{\overline {703}}$, so we reach the seemingly absurd conclusion that $f(x)=x^{3}$ is not differentiable where $x>0.339{\overline {703}}$.

This complication is not present in the standard definition of derivative, where the remainder terms go away in the limit and we are just left with $f'(x)=3x^{2}$. It's not completely clear whether or not this complication is a problem. Those pesky terms are measuring something real, which calculus is ignoring. They are measuring the difference between two different methods of finding the slope of the tangent line to a real curve.