The difference between $\Delta x$, $\delta x$ and $dx$ $\Delta x$, $\delta x$ and $dx$ are used when talking about slopes and derivatives. But I don't know what the exact difference is between them.

Let $y=f(x)$.
$\Delta x$ and $\delta x$ both denote the change in $x$ (or increment of $x$). Some books prefer to use capital delta $\Delta$ and some lowercase delta $\delta$. The change can be small or large, but often we talk about the case that $\Delta x$ is small and especially $\Delta x\to 0$.
In mathematics, $dx$ is another independent variable which can assume any number.
$$-\infty<dx<\infty.$$
$df$ is a function of two variable $x$ and $dx$. Its value is denoted by $dy$ and
$$dy=f'(x)dx$$
Therefore,$dx$ does not need to be small. Often in calculus, it is said that $dx=\Delta x$, but there is a difference between them. While $\Delta x$ must be small enough such that $x+\Delta x$ lies within the domain of $f$, there is no restriction on $dx$.
In an old-fashioned approach which is still vastly used in physics, $dx$ is called an infinitesimal (or infinitely small change in $x$).
In calculus, use of infinitesimals is sometimes beneficial especially in integration applications (for example to derive the formula of the arc length or areas of surfaces of revolution, etc.) as we do not need to go through limit of sums process.
Unfortunately there are many answers to this question which are completely off-base. And my original answer was deleted, so now I have to add the following reference from Thomas's calculus, 3rd edition.


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*The notation for partial differentiation is $\partial x$ (and not $\delta x$).


*$dx$ is not the limit of $\Delta x$. The limit of $\Delta x$ is zero when $\Delta x\to 0$.


*I did not talk about non-standard analysis.
Here is the link for those who want to study more (George B Thomas, Calculus and Analytic Geometry, 3rd edition, p. 82):
