TITLE: Rigorously, what's happening when I treat $\frac{dy}{dx}$ as a fraction?
QUESTION [6 upvotes]: Consider a very simple differential equation (assuming $y$ is an invertible, continuous function and considering only real numbers, for simplicity):
$$\frac{dy}{dx}=y$$
If I were to solve this, I would do something like:
$$dy=dx\cdot y \implies \frac{1}{y}\cdot dy = dx$$
Then I would integrate to arrive to a solution. Of course, this works, but why? $\frac{dy}{dx}$ is not a fraction, even if it sometimes acts like one. So what are the missing steps that allow you to treat it as such? I seem to remember an explanation but I can't find it anywhere, so I'm asking here. Thanks!

REPLY [5 votes]: Everything that looks like a fraction manipulation of things is actually an entire theorem about derivatives that we are applying, often nontrivially.

The main thing behind doing $~\frac{dy}{dt}~\frac{dt}{dx}=\frac{dy}{dx}~,$ we are imposing the Chain Rule.
If we take $~\frac{1}{dy/dx}=\frac{dx}{dy}~,$ then we are using the Inverse Function Theorem.
Also we have to be pretty careful how we plug values into our functions. 
$(a)~$If we are integrating $~f(y(x))\frac{dy}{dx}dx~$ over some interval, and in the next step we write it as a integral of $~f(y)dy~$ over some other interval $($i.e., $u$-substitution$)$, then we are using the Fundamental Theorem of Calculus. 
$(b)~$Even Separation of Variables, from differential equations, is not given by manipulating differentials as fractions, but rather through clever use of the Chain Rule.

At no point should we pretend that $~dy/dx~$ is a fraction. What we can do is use our intuition about fractions to guide what we do when working with derivatives. Since a fraction does $~\frac{1}{a/b}=\frac{b}{a}~,$ it may be seems that "Maybe there is an analogous result about derivatives", and then find a theorem that says that $~\frac{1}{dy/dx}=\frac{dx}{dy}~,$ and see how it is precisely used (because we have to be careful how we plug stuff in). Every time we do a fraction-like manipulation of derivatives, we should think "This is because of Such-n-Such Theorem", if we don't know what theorem we are using to do the manipulation, then we shouldn't do it. At least until we find the theorem the says we can do it.
The reason why $~dy/dx~$ behave "Fraction-like", is because they are limits of things that are fractions. This does not imply that $~dy/dx~$ is a fraction, but rather that sometimes the manipulations that we do to fractions survive through the limiting process, giving us one of these main theorems. But each time, it is a nontrivial result and it is good to have all of them in the back of our mind, and to ponder about them, or whatever.
One main problem area in viewing these as fractions is when we go into higher dimensions. Here, the fraction analog breaks down almost completely. What we find is that derivatives behave more like operations from linear algebra that, in one dimension, can seem to be fraction-like, but are not so in general.

You may also see the answers from the following links:
$1.~~$ When can we use $dy/dx$ as a fraction ?
$2.~~$ How misleading is it to regard $dy/dx$ as a fraction ?
$3.~~$ Why Do People Treat $dy/dx$ as a Fraction ?
$4.~~$ Can I treat the notation $dy/dx$ as a fraction ?
$5.~~$ When not to treat dy/dx as a fraction in single-variable calculus?
$6.~~$ Why do people say dy/dx is not a fraction, but then use it as one when doing the chain rule?
$7.~~$ What is wrong with treating $\dfrac {dy}{dx}$ as a fraction?
$8.~~$ When can we not treat differentials as fractions? And when is it perfectly OK?
$9.~~$ Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
and so on in available in the internet.