Sometimes I see that, during a $u$-substitution in an integral (which is something I hardly understand), once $u$ is chosen it is differentiated to obtain $du/dx$. I understand this. After, however, people solve for $dx$ as though it is a regular variable, and treat $du/dx$ as though it is a regular fraction. My googling has told me that $du/dx$ isn't a fraction and that $dx$ or $du$ aren't really regular variables, as one is dependent on the other. So why can they behave like what they are not?
For example, if $u$ is $x^2$, then $du/dx$ ends up being $2x$, and so $du = 2x dx$ and $dx = du/2x$. What my question is, why is this allowed?
(Also, sorry that I haven't properly formatted the previous equations using latex or whatever it should be, as i don't know about any of that right now.)
