Leibniz notation - how to get $dx$ out of a derivative $v = \frac{dx}{dt}$ I know that velocity equals $v = \frac{dx}{dt}$ which is writen in Leibniz notation. How can i get $dx$ out of it in a proper way? I don't like it when people say that i should just multiply denominator $dt$ with $v$ like this: 
$$
\begin{split}
v &= \frac{dx}{dt}\\
v\, dt &= dx \longrightarrow \boxed{dx = v \, dt}
\end{split}
$$
I think that there must be some rules behind this which aren't mentioned most of the time. So what are these rules? How can i get same result in a proper mathematical way? Could you recommend any good book which focuses only on Leibniz notation and its tricks.
 A: Not one of my best answers, but it is what it is...
It is notation, just like much of math.  Using prime notation, we can define this quantity called the differential.  For the below example, the differential is $dx$:
$$dx = f'(t) dt$$
This is why you will see confusion when performing separation of variables.  Really, the $dx$, $dt$, or whatever other variable at the end of an integral is a differential.  That is why we can simplify:
$$\frac{dx}{dt}=xt$$
$$\frac{1}{x}\frac{dx}{dt}=t$$
$$\int\frac{1}{x}\frac{dx}{dt}dt=\int t\;dt$$
Note that we're realy integrating both sides with respect to t!  However, $\frac{dx}{dt}dt = dx$, based on the definition of the differential.
So, this becomes:
$$\int\frac{1}{x}dx=\int t\;dt$$
A: We know that velocity and position functions are functions of time such as $x(t)$ and $v(t)$. By taking an infinitesimal increment of position with respect to an infinitesimal increment of  time we get
$$d\ x(t)=\frac {d\ x(t)}{d\ t}d\ t$$
and by definition of velocity we know that
$$v(t)=\frac {d\ x(t)}{d\ t}$$
which reveals
$$x(t)=\frac {d\ x(t)}{d\ t}d\ t=v(t)\ d\ t$$
