A basic differential inside an integral If I am not mistaken a differential is defined like this (please correct me if I am in any way wrong):

If we take infinitesimally small part of a function changes become smaller and we rename them accordingly $\Delta x\rightarrow dx$ and $\Delta y\rightarrow dy$. Than we can define a rate of change (linear coefficient) $k$ like 
  $$
\begin{split}
\underbrace{k=\frac{\Delta y}{\Delta x}}_{\text{for bigger changes}} \underrightarrow{\scriptsize~~\text{as changes get smaller}~~~}k=\frac{dy}{dx}\\
\end{split}
$$
  But we also know that $k$ equals a derivative $\frac{dx}{dy}$ hence:
  $$
\begin{split}
\underbrace{\frac{dx}{dy}}_\text{derivative - do not 
separate}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!=\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\overbrace{\frac{dy}{dx}}^\text{rate of change - can be separated}\\
\end{split}
$$
  And now we separate rates of change while we must not separate derivative right?
  $$
\boxed{dy=\frac{dy}{dx}\,dx}
$$

Now comes my real question. If I have a definite integral for example:
$$
\int\limits_0^t\frac{dx}{dt}a \!\!\!\!\!\!\underbrace{dt}_{\text{belongs to }\int\limits_0^t}
$$
I can see that last $dt$ is part of an integral notation and cant just be moved right? So why do some authors treat it like it is an ordinary fraction (cancel out the $dt$) and use this assumption to change even integration limits like this:
$$
\int\limits_0^t\frac{dx}{dt}a\,dt=\int\limits_0^x a\,dx
$$
I would like to clear this up in my head once and for all.
 A: Well , we have chain rule in differentiation and its  counterpart integration by substitution.
These are theorems that require proof.So what you do here is not cancelling in algebra.But in 
effect , it seems like you just cancel them which confuses you..
A: I had the same problem with this when I began learning calculus. I was told that the $dt$ in an integral was simply notational and I couldn't understand how my teachers told me that they "cancel out". Basically, the thing which you put in a box, $dy = \frac{dy}{dx}\, dx$, only really makes sense in terms of integrals.
What I mean is that
$\int dy = \int\frac{dy}{dx}\, dx$
by the chain rule, so as a short hand we sometimes write
$dy = \frac{dy}{dx}\, dx$
with the understanding that we have not defined some algebra of infitesimal quantities - we're just using a shorthand.

Also, the derivative is more commonly defined as
$\frac{dy}{dx} = \lim_{\delta x\to0} \frac{y(x + \delta x) - y(x)}{\delta x}$
which, if you think about it, is intuitively similar to what you wrote.
