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When we write $dx^2$, do we actually mean $$d(x^2)$$ (the change in respect to $x^2$), or $$(dx)^2$$ ($dx$ squared)?

gldanoob
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  • Do you mean $\frac{d^{2}y}{dx^{2}}$? – DDS Jul 08 '19 at 04:21
  • $\dfrac d {dx} \dfrac d {dx} y = (\dfrac d {dx})^2 y = \dfrac {d^2} {dx^2} y$ – J. W. Tanner Jul 08 '19 at 04:24
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    It's better to put such things in context. Where exactly did you see this? As part of what expression? – Michael Rybkin Jul 08 '19 at 04:24
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    I don’t think this question is very meaningful without further context. Each person can give their opinion about which usage is more common, but given that my judgment is the exact opposite of the current answer’s, that seems an unproductive exercise. The onus is on the OP to provide more details about the question: how does it arise? – Erick Wong Jul 08 '19 at 04:25
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    Possible duplicate of What does $\ dx^2$ mean? – nmasanta Jul 08 '19 at 07:31
  • my question is about what do we mean to square when we write $dx^2$ (Are we just squaring the $x$ or the entire $dx$ itself). – gldanoob Jul 08 '19 at 10:12
  • For example, when we are using expressions to write the derivative of $y = x^2$, if we want to replace the $y$ with $x^2$ in $\frac{dy}{dx}$, can we write $\frac{dx^2}{dx}$? Or people will interpret $dx^2$ as $(dx)^2$ so I cannot do that? – gldanoob Jul 08 '19 at 10:16
  • You usually write it like this: $$\frac{d}{dx}\left(x^2\right).$$ It means the same thing as $$y'=\left(x^2\right)'.$$. – Michael Rybkin Jul 08 '19 at 12:11
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    I don't see how it's "unclear what he's asking". Seems perfectly clear: Does the notation $dx^2$ refer to $(dx)^2$ or $d(x^2)$? Of course the answer is sometimes the first and sometimes the second. There should be an answer explaining this - so we should reopen... – David C. Ullrich Jul 08 '19 at 14:35
  • @DavidC.Ullrich Yeah that's exactly what I'm trying to ask – gldanoob Jul 09 '19 at 01:54

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The usual notation in Stieltjes integrals for differentiable $f(x)$ is $$ df(x) = f'(x) dx $$ so in your case $$ d\left(x^2\right) = 2xdx $$

gt6989b
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