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.
 A: $\Delta x$, is used when you are referring to "large" changes, e.g. the change from 5 to 9.
$\partial x$ is used to denote partial derivative when you have a multivariate function (e.g. one with x,y,w, instead of just x alone).
$dx$ is used to denote the derivative when you have a univariate function (when you just have x and there is no confusion).
A: There are several answers to similar/the same questions:


*

*Given $z=f(x,y)$, what's the difference between $\frac{dz}{dx}$ and $ \frac{\partial f}{\partial x}$?

*What is the difference between $d$ and $\partial$?
But the answer from Tom Au also puts it in a nutshell.
A: A 100-words' answer is not required to explain this (as other answers).
See this answer is Quora : What is the difference between dx and Δx?

$ \Delta x$ is a small change(in the context you have used it in) in
$x$.
$ dx$ is a vanishingly small change in $x$.
$dx $ is
obtained when $ \Delta x$ tends to zero.
​   ​
​​Look at the width of
the rectangles.
Their size gradually decreases as you can see when you move through
the four graphs. Only when the rectangle width becomes vanishingly
small, the width is called $ dx$.

I think this is the best explanation so far.
A: $\Delta x$ is about a secant line, a line between two points representing the rate of change between those two points. That's a "differential" (between the two points).
$dx$ is about a tangent line to one point, representing an instantaneous rate of change. That makes it a "derivative."
$\delta x$ is about a tangent line to a partial derivative. That's a rate of change or derivative in one direction, holding a number of other directions constant.
A: 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.


*

*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):
