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.