TITLE: What does $dx$ mean?
QUESTION [51 upvotes]: $dx$ 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.

REPLY [3 votes]: I have a relevant blog post for those comfortable with multivariate calculus, found here.
Though treating $\mathrm{d}x$ as simply "that thing" works, there is hidden meaning behind it. To be clear, in this case, we are using $x$ to refer to the identity function $x(t)=t$, where $t\in\mathbb{R}$.
Imagine an arrow tangent to the real line. This can be represented by a vector (magnitude and direction) and a real number (position). We call such a vector a tangent vector. If the vector is $v$ and the position is $p$, then we denote the corresponding tangent vector* by $v_p$.
Mathematicians define $\mathrm{d}x$ as a type of function, called a differential $1$-form, that takes in a position $p$ and outputs a so-called "dual" tangent vector, or cotangent vector, which we call $\mathrm{d}x_p$. This $\mathrm{d}x_p$ is, somewhat confusingly, a function on the tangent vectors positioned at $p$. In the case of real numbers, though, the expression for $\mathrm{d}x_p$ is easy: $$\mathrm{d}x_p(v_p)=v.$$
Though many people will prefer to work with the (arguably more rudimentary) notion of "infinitesimals", differential forms have many advantages over the "old way". For example, change of variables, commonly known as $u$-substitution, has a simple formula in terms of something called the "pullback". As another example, when we move on to calculus on spaces that aren't strictly Euclidean, these differential forms give powerful information about the space itself, such as (in some sense) how many holes it has.
(* For the professionals, what I am meaning here is that $v_p=v\left.\frac{d}{dx}\right|_p$. Feel free to heckle in the comments.)