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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.

gldanoob
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    The confusion in the prior question was due to the lack of context, but you still provide no context. Usually if I saw a thing like this I would assume that d$(x^2)$ was intended but perhaps you are thinking of some specific situation in which this was not intended, as in the denominator of the second derivative. As a general rule: if you feel that a notation is ambiguous, add parentheses (or text) to clarify. – lulu Jan 20 '20 at 11:08
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    It depends. There is no absolute answer. Mathematicians are human, too. If the meaning is obvious from context then it will be whichever one fits better. I'm not sure why you want to insist that $dx^2$ has to have a definitive meaning. But the two expressions with the parentheses made clear ($(dx)^2$ vs $d(x^2)$) are distinct from each other. – Ninad Munshi Jan 20 '20 at 11:08
  • See Differentiation : notation – Mauro ALLEGRANZA Jan 20 '20 at 11:45
  • See Differentiation : notation: $\dfrac {d^2y}{dx^2}$ is the second derivative of $y$ with respect to $x$ – Mauro ALLEGRANZA Jan 20 '20 at 11:46
  • And $d^2 y=d(dy)$ is the second order difefrential, i.e. the iteration of the operation of "differentiation". – Mauro ALLEGRANZA Jan 20 '20 at 11:49
  • There is no conflict in the previous question if you edit it so that it becomes clearer. If you think the question is still the same but there was a misunderstanding, edit it, don't post another question. – Simply Beautiful Art Jan 20 '20 at 14:43

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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.

GEdgar
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