I've already tried to ask this question (Is writing $dx^2$ same as writing $d(x^2)$ in calculus), but people seems to misunderstand what I was trying to express, so I decided to ask it again.
My question is, if one writes $\mathrm{d}x^2$ (which is kind of a differential expression), should it be read as $$(\mathrm{d}x)^2$$ (differential of x, but squared) or $$\mathrm{d}(x^2)$$ (differential of $x^2$)?
The question can also be asked in a different way: if I want to express $(\mathrm{d}x)^2$, do I have to explicitly write the parentheses or just write $\mathrm{d}x^2$?
Keep in mind that I'm not asking about the meaning of the differential, but instead the notation related to it.
When used for second derivative $$ \frac{d^2y}{dx^2} $$ just recognize it as "second derivative". Maybe once in your life look up the fact that it abbreviates $$ \left(\frac{d}{dx}\right)^2 y $$ But then always think merely "second derivative". Do not try to make sense of the "$dx^2$" by itself.
Another use is for a certain type of element of arc length: $$ ds^2 = E\;dx^2 + F\; dx\;dy + G \; dy^2 $$ There are thought of as $(ds)^2$ and so on.