From a mathematician's point of view, what is the purpose of '$dx$' in $\int f(x)\ dx$?

I've done a bit of searching and found a fairly well written explanation, but at the end, the author noted that this explanation seems to work fine for a physicist's purposes - but a mathematician would groan at it due to oversimplifications or inaccuracies.

Since I first posted this paper, two different people have emailed me to tell me that Real Mathematicians don't do this. Playing with dx in the ways described in this paper is apparently one of those smarmy tricks that physicists use to give headaches to mathematicians.

http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/dx.html

I was also confused when reading it because my Calculus prof last semester said that the chain rule

$$\frac{dy}{dz}=\frac{dy}{dx}\cdot \frac{dx}{dz}$$

cannot be treated as fractions, despite the fact that they look like it, and the $dx$ would not cancel out between the two since you can't do that with differentials. But this article said just the opposite.

I have heard from several mathematicians that one thing that they didn't like about physics was the "abuse" of mathematics. Physicists are not necessarily bad mathematicians, they (IMO) just use math differently. For them, the precise nature isn't (always) at the core of what they are doing. Since the physicist works with the "real world", he/she doesn't often work with approximations and therefore is not passionately interested in the exact nature of equations and such (hopefully I didn't offend anyone).

I see on the site that you give reference to, that the person thinks about $dx$ as a finite quantity. He talks for example about $x + dx$ as being a real number. This is not necessarily a bad way to think about it, but again it isn't the exact definition.

About the integral: When we write for example $$ \int_a^b f(x)\; dx $$ we have defined this symbol as is. We haven't defined the symbol as made up of different individual component that can we written on their own. So even though $f(x)$ in the expression indeed does have meaning on its own, we are not allowed to remove it and just write: $$ \int_a^b dx. $$ Now, someone might actually use this notation from time to time, but in "standard basic calculus" this hasn't been defined, and so it doesn't have any meaning. (It doesn't mean that we can think about it in a certain way).

Now all that said, this doesn't mean that there isn't a rationale behind the choice of the different components of the symbol. So, of course, the $f(x)$ is the function. The $a$ and the $b$ are the limits of the integral and the $dx$ tells us to treat $x$ as the variable (we say that we integrate with respect to $x$). Why is this important? As mentioned in another answer, if you consider the integral: $$ \int_{a}^{b} x^2y. $$ you wouldn't know if $x$ is the variable of $y$ is the variable (or if both are constants). So that is why we like to "add" the $dx$ or $dy$ to the end.

This isn't the only reason. The definite integral $$ \int_a^b f(x)\; dx $$ is defined as a limit: $$ \lim_{n\to \infty}\sum_{i=1}^{n} f(x_i)\Delta x. $$ Here $\Delta x$ is a finite number defined as $\Delta x = \frac{b - a}{n}$. And taking the limit $n\to \infty$ we have $\Delta x \to 0$. So we "think" of $dx$ as this infinitely small number. We can "think" of the $\Delta$ as becomming the $d$, even though the $dx$ can't be written on its own.

The same applies to the derivative $$ \frac{dx}{dy}. $$ Here again we have just defined this one symbol where the parts of the symbol ($dx$ and $dy$) are not well defined on their own.

All this said, one can actually make a meaning (i.e. define) of $dx$, but when we are just taking about basic calculus, we don't usually do that.

Also note that some mathematicians will just write $\int f(x)$ when they mean $\int f(x) \; dx$ because it is "obvious" that we are integrating with respect to $x$.

You can find more things about the $dx$ in this question: What is $dx$ in integration?

A possibility is to see $dx$ as a differential form. Then, the integration for differential forms is well defined.