Manipulation of calculus notation I'm trying to get my calculus back up to scratch after not using it for 20 odd years. During my research, I've just seen this on https://physics.info/kinematics-calculus/:
$$a = \frac{dv}{dt}$$
$$dv = a\ dt$$
$$\int_{v_0}^v dv = \int_0^{\Delta t} a\ dt$$
Is this a valid manipulation of the calculous notation? My understanding is that the d/dt syntax is just notational, so I'm surprised to see it being treated as if the dt and dv where just algebraic variables that can be manipulated in this way.
Specifically, multiply both side by $dt$ is a surprise. Supposing for example we had this:
$$a = \frac{d}{dt}\ f(x)$$
Now, if I do the same thing as above, I get this:
$$d\ f(x) = a\ dt$$
Which looks like nonsense.

It is fine, mostly. Infinitesmals are really defined under any number system, even when one considers proffs in calculaus/analysis, one deals with epsilon-delta proofs that skirt arounf the issue of infinitesmals.
Mathematically however, that is treating those quantities as manipulated 'variables' one could write $dy=f'(x)dx$ as the mathematical definisiont of the infinitesmal $dy$ as the differential is the infinitesmal difference between two successive values of a fucntion.
What is also required is a consideration of the idea of a separable equation, namely, that $dy$ and $dx$ are just notation here. And, more importantly, they are not necessary for separation of variables.
A separable equation is of the form
$$g(y)y′=f(x)$$
