# What does $dx$ mean?

$dx$ 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.

The formal definition of an expression such as
$$\int_0^1 x^2\,dx$$
will depend on the setting. This is because there is not just one “theory of integration” – there are several different theories in different areas.

I like the presentation at the beginning of this note by Terence Tao. The key point is that there are really at least three different viewpoints on integration in elementary calculus:

• Indefinite integration, which computes antiderivatives

• An “unsigned definite integral” for finding areas under curves and masses of objects

• A “signed definite integral” for computing work and other “net change” calculations.

The value of an expression such as $\int_0^1 x^2\,dx$ comes out the same under all these interpretations, of course.

In more general settings, the three interpretations generalize in different ways, so that the “dx” comes to mean different things. In the setting of measure theory, “dx” is interpreted as a measure; in the context of differential geometry, it is interpreted as a 1-form.

But, for the purposes of elementary calculus, the only role of the “dx” is to tell which variable is the variable of integration. In other words, it lets us distinguish
$$\int_0^1 uv\,du = v/2$$
from
$$\int_0^1 uv\,dv = u/2$$