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I'm trying to gain a more conceptual understanding of derivatives and would appreciate your feedback on this.

Say I have a quantity, $x$, at time $t$. Now $x$ moves to a different location $x'$ in time $t'$ = $t + \Delta t$.

Where I get confused is when we start talking about shrinking $\Delta t$ down to zero. I keep seeing people say that it represents an infinitesimal quantity, which confuses me even more. Similarly, people will say it "simply" represents a very small quantity.

I get that much but where I get lost is how small does $\Delta t$ have to be before we start treating it as $dt$ and not $\Delta t$?

In other words, is it correct to simply substitute numbers in to a quantity like $dt$? Could I say that at a certain instant in time, $dt$ = 4 seconds?

I've seen this done before in a few books and well, frankly it irritates me because I'm seeing the $d$ operator used in many different contexts. Some are saying you can substitute numbers in for something like $dt$ and others say no.

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$\Delta t$ is used in taking the limit to arrive at the derivative (or the integral).

For example:

$\frac{df(x)}{dt} = lim \frac {(f(x+\Delta t)-f(x))}{\Delta t}$ as $\Delta t$ tends to zero

so $dt$ is used after the limit has been taken whereas $\Delta t$ is used before or during the limiting process. So its really not a question of how small $\Delta t$ has to be before it becomes $dt$ as long as a limit can be found as it approaches zero. Then the suggestive symbol $\frac{df(x)}{dt}$ can be used for that limit. In that sense the numerator and denominator are not quantities.

$\Delta t$ is sometimes used in other contexts where it is just a small increment of something--the required smallness being defined by the user. For example in digital quantization.

I think the term 'infinitesimal' is considered somewhat controversial by mathematicians where the emphasis is on the limiting process without getting into what an infinitesimal is. Non Standard Analysis (Abraham Robinson) and smooth infinitesimal analysis (J.L. Bell) have been invented to deal with them.

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