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$dx$ appears in differential equations, such us derivatives and integrals.

For example, a function $f(x)$ its first derivative is $\dfrac{d}{dx}f(x)$ and its integral $\displaystyle\int f(x)dx$. But I don't really understand what $dx$ is.

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good question, but you can find lot about it in web as well ! – Theorem May 9 '12 at 21:39
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Although the title is not exactly the same as your question, I believe your question is answered quite thoroughly in this similar post: math.stackexchange.com/questions/21199/is-dy-dx-not-a-ratio – Michael Boratko May 9 '12 at 21:39
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@Garmen: Please take a look at my comments under the answer you accepted. That answer is quite misleading; one of the limits is wrong, and the concept "infinitesimally small" is being used informally without a definition. While there is an interesting branch of mathematics called non-standard analysis that defines infinitesimal quantities, standard analysis (which is presumably what you're asking about) has no such concept. – joriki May 10 '12 at 21:42
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3 Answers

up vote 6 down vote accepted

Formally, $dx$ does not mean anything. It's just a syntactical device to tell you the variable to differentiate with respect to or the integration variable.

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IMHO this is just one, very unenlightening way to look at it. – Michael May 10 '12 at 10:01
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@Michael, sorry, can't tell my math-for-poets classes that it's a differential one-form; I can (and do) give them lhf's answer, and they get some mileage out of it. – Gerry Myerson May 10 '12 at 13:18
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your answer is not clear and not answer the question, because if just we consider tow function $F(x,y)$ and $G(x,y)$ defined on $\mathbb{R}^2$, what would be the meaning of $F(x,y)dx-G(x,y)dy=0$ by the use of your definition !!!? – Abdelmajid Khadari May 11 '12 at 0:48
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your answer is not correct. If you see a equation with dx and dy then they mean something and it is not simply a part of a derivative or an integral description. And you can't always simply split them to solve this equation – Gargo May 16 '12 at 7:24
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@GerryMyerson: I've never taught math-for-poets, but I wonder if teaching such people formal laws for manipulating symbols, without telling them about the meaning of it is more valuable than trying to explain in poetic terms what a differential one form is. (Maybe something like infinitesimal coordinates?) – Michael May 23 '12 at 9:44
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up vote 6 down vote

As Silvanus Thompson put it in his book Calculus made easy: dx means "a little bit of x".

If that is not satisfying, there are various more precise explanations. One of them is: dx is a differential one-form.

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up vote 2 down vote

$dx$ means a very very small quantity, $dx=x_2-x_1$ where $x_1$ and $x_2$ very very near to $x$ (in geometry a very small distance), when you derive $\frac{d}{dx}f(x)$ it means you calculate the propinquity of $df(x)$ and $dx$, when you integrate, the sign $\int$ means a continuous sum, so $\int f(x) dx$ means a continuous sum of all the quantities $f(x) dx$ (geometrically very very small rectangles), in graduate language $dx$ is a linear map (differential form).

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If I didn't already know what $dx$ was, I don't think this answer would clear anything up for me. – Gerry Myerson May 10 '12 at 23:48

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