difference between tending to zero and close to zero

Question

In calculus we are taught that the derivative of function $y$ with respect to $x$ is defined as "the quantity which $\dfrac{\bigtriangleup{y}}{\bigtriangleup{x}}$ tends to when $\bigtriangleup{x}$ tends to zero"

Can we also define it as "the ratio of $\dfrac{\bigtriangleup{y}}{\bigtriangleup{x}}$ when $\bigtriangleup{x}$ is a quantity very close to zero"?

Answer 1

Let $y = x^2$. Then $\dfrac{dy}{dx} = 2x$ for the usual reasons

$\lim_{\Delta x \rightarrow 0} \dfrac {\Delta y}{\Delta x} = \lim \dfrac {[x^2 + 2x\Delta x + (\Delta x)^2] -[x^2]}{\Delta x} = \lim (2x + \Delta x)= 2x$.

But if we chose to say that the derivative is a value where $\Delta x$ is a number "very near zero" (I'm going to ignore the difficulty of how one would define such an ambiguous expression) so that $\Delta x = \delta >0$ but $\delta$ is "small".

Then $\dfrac {\Delta y}{\Delta x} = \dfrac {(x+\delta)^2 - x^2}{\delta} \dfrac {x^2 + 2x\delta + \delta^2 }{\delta} = 2x + \delta \ne 2x$.

So, no, that is a different answer. You can't say $\Delta x$ isn't zero at the beginning and then say "well $\Delta x$ very close to $0$ so we can ignore it" in the end.

Answer 2

You can consider the derivative to be the shadow of the ratio $\frac{\Delta y}{\Delta x}$ when $\Delta x$ is infinitesimal. Alternatively, you can exploit the kind of relation Leibniz had in mind, which was a relation of infinite proximity. Leibniz wrote in his articles that when he speaks of equality, he does not mean strict equality but rather equality up to a negligible term.

Leibniz did not distinguish in notation between strict equality and equality up to a neglibigle infinitesimal, but he did use an alternative notation ${}_{\ulcorner\!\urcorner}\,$ for such a relation so it may be instructive to express this thought as follows. If $y=f(x)$ then one can write that $f'(x)\;{}_{\ulcorner\!\urcorner}\,\frac{\Delta y}{\Delta x}$ while understanding that $f'(x)$ has to be real (or, as Leibniz would have put it, "assignable").

Shadow is the same as the standard part.

Incidentally, Leibniz's remarks about a notion of equality "up to a negligible term" show that Bishop Berkeley's criticism of the calculus had no merit; see this article for the details.