# 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.

• Well, $\delta x$ means different things depending on the context. For example, it has a particular meaning in variational calculus, and a completely different one in functional calculus... – Alex Nelson Dec 27 '13 at 15:20

$$\Delta x$$ 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.

• Note that in physics $\Delta x$ is also used informally to mean "small increment in $x$". And $\delta x$ can also mean a variation of a function, in the calculus of variations. I guess my point is that calculus notation is still far from standardized, and you will often need to infer the meaning of notation from the context. – user7530 Dec 27 '13 at 16:50
• @user7530 My experience is that $\Delta x$ denotes the difference, or change of $x$; no matter how big or small. For example $\Delta W = F \times \Delta x$. Then it is common to write $$\lim_{\Delta x \to 0} \frac{\Delta W}{\Delta x} = \frac{\operatorname{d}\!W}{\operatorname{d}\!x}$$ – Fly by Night Dec 27 '13 at 17:19
• @FlybyNight Yes I've seen that. I've also seen several calculations where $(\Delta x)^2\approx 0$. shrug – user7530 Dec 27 '13 at 18:49
• @user7530 I'd be interested to this in context. Do you have a link? – Fly by Night Dec 27 '13 at 19:15
• The reason for that is, that if $\Delta x$ is "small", then $(\Delta x)^2$ is even smaller – Fakemistake Dec 30 '18 at 10:19

$\Delta x$, 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).

There are several answers to similar/the same questions:

But the answer from Tom Au also puts it in a nutshell.

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.

$$∆x~$$ 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