# Trying to understand the difference between $\Delta t$ and $dt$ [duplicate]

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.

• Possible duplicate of Difference between $\Delta$, $d$ and $\delta$ – AccidentalFourierTransform Jun 2 '18 at 20:42
• Under a sufficient magnification curves appear as straight lines. You can start treating $\Delta t$ as $\text{d}t$ when everything around becomes linear. In other words, when $(\Delta t)^2$ becomes small enough to be neglected compared to $\Delta t$. – safesphere Jun 3 '18 at 0:28
• – Kyle Kanos Jun 3 '18 at 0:29

$\Delta t$ is used in taking the limit to arrive at the derivative (or the integral).
$\frac{df(x)}{dt} = lim \frac {(f(x+\Delta t)-f(x))}{\Delta t}$ 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.