The difference between $\Delta x$, $\delta x$ and $dx$
Matthew Harrington 
$\Delta x$, $\delta x$ and $dx$ are used when talking about slopes and derivatives. But I don't know what the exact difference is between them.
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$\begingroup$$\Delta x$ is about a secant line, a line between two points representing the rate of change between those two points. That's a "differential" (between the two points).
$dx$ is about a tangent line to one point, representing an instantaneous rate of change. That makes it a "derivative."
$\delta x$ is about a tangent line to a partial derivative. That's a rate of change or derivative in one direction, holding a number of other directions constant.
$\endgroup$ 5 $\begingroup$$\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).
$\endgroup$ $\begingroup$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.
$\endgroup$ $\begingroup$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.
$\endgroup$ $\begingroup$$∆x~$ is small change in $~x~$.
$~dx~$ is small part of $~x~$ but represents independent change. & $~\frac{dy}{dx}~$ means slope of tangent at a point where it touches to the curve $~\frac{∆y}{∆x}~$ s the slope through two points.
We say $~∆x~$ tends to zero .
It becomes $~\frac{dy}{dx}~$ which is slope of tangent at a point (reason of why $~∆x~$ tends to zero).
That is it
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