If I have the derivative, $${df \over dx} = f'(x)$$
To integrate, is it correct to separate the $df$ and $dx$ differentials: $${df \over dx} \hphantom{.} = \hphantom{....}f'(x)\hphantom{.}$$ $$\hphantom{...} df \hphantom{.} = \hphantom{....}f'(x) dx$$ $$\int \Big{[} df \Big{]} = \int \Big{[} f'(x) dx \Big{]}$$
Or can it only be done like this:
$${df \over dx} = f'(x)$$ $$\int \Big{[}{df \over dx}\Big{]} dx = \int \Big{[}{f'(x)}\Big{]} dx$$
(Or are both correct?) I am asking because recently I learned if you have a total differential $df = y\hphantom{.} dx + x\hphantom{.} dy$, you can't just integrate like this $\int df = \int y\hphantom{.} dx + \int x\hphantom{.} dy$ as this would get you $f(x,y) = 2xy$ instead of $xy$.
Instead, you have to integrate like $(\int y dx + \int x\hphantom{.} {dy \over dx}dx) \rightarrow xy + c(y) + 0 = xy + c$. And I'm wondering if that also applies to the $df$ on the LHS, where you would have to integrate with ${df \over dx}{dx}$ instead of just $df$.
Thanks.
Its just a notation, you can use $df$ to mean $f'(x)dx$ if you like, but its not an standard notation for a basic calculus course. Some areas where this notation could appear as a standard notation:
In differential geometry, $df$ represent what is called a differential form or the differential of a smooth map.
In measure theory, if $f$ is a function of bounded variation between real or complex finite dimensional vector spaces, then $df$ represents a complex measure (this notation is commonly used in probability theory).
In an hyperreal setting $df$ represents an infinitesimal (indeed, in this setting, the expression $df/dx$ is a quotient of infinitesimal numbers, where $dx\neq 0$ and $df$ is just a notation for the infinitesimal $f'(x)dx$).
There are other places where this notation can appear, but they are rarer (in general something like this appear in alternative theories for differential geometry or calculus).