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I have never seen this notation, but I think that it follows in a similar vein for function notation. So if $y=f(x)$, then $y''=f''(x)$.

Then by that, can we say that

$$f^{(n)}(x)=y^{(n)}$$

Eleven-Eleven
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    It is widely used, e.g. see Lagrange's notation. – Bill Dubuque Jun 04 '18 at 18:30
  • If in doubt, when you use it explain it the first time. – Gerald Edgar Jun 04 '18 at 20:06
  • @Number actually it is Euler's notation – Michael Bächtold Jun 05 '18 at 05:15
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    It's common. The rationale is that it is hard to count 4 or 5 dashes. I have not seen any derivatives above 5 though, in the sciences (and even 4 or 5 are rare). I am used to seeing the 4 or 5 in Roman numerals, occasionally lower case. – guest Jun 05 '18 at 09:47
  • @MichaelBächtold My comment simply quotes the name of a section of the Wikipedia page. It was not meant to imply anything about the history. Do you know if Euler used it for general $n$ as in the final equation in the OP? (I don't see that in the link you gave). – Bill Dubuque Jun 05 '18 at 15:51
  • @guest: I don't know about recent uses of Roman numerals, but I've seen quite a few such uses in old (1800s) literature, such as those I cited here. I suspect this was done to distinguish between a superscript used for the order of differentiation and a superscript used for exponentiation. – Dave L Renfro Jun 05 '18 at 16:12
  • I don't have the sort of familiarity as you, but know I have seen it. Am out and about but can check my tiny library later. however, even the Wiki article (linked above) shows the Roman numerals. Even before the Arabic ones (i.e. somewhat more common.) – guest Jun 05 '18 at 16:16
  • @Number sorry if my comment came off as a critique. I just wanted to use the occasion to point out the widespread wrong attribution. Concerning general $n$ in Euler's notation: I don't know. – Michael Bächtold Jun 05 '18 at 20:18

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It can be used, but it's a bad piece notation, just as $y'$ is. I have two reasons for saying that.

  1. $y'$ doesn't indicate with respect to which variable you differentiate. So for example, if $y=t^2$ and $t=e^x$, what should $y'$ denote? Is it $dy/dt$ or $dy/dx$? I see students getting confused by this when they try to derive with the chain rule using the prime notation.

  2. Even if you object and say: in my context $y'$ will always denote derivative with respect to $x$, I still consider it a bad notation since it's the same notation we use for $f'$. Hence it suggest to students that $y$ and $f$ are the same type of objects.

The only justification I see for using it is laziness and tradition.

Michael Bächtold
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    The prime notation, $x^{\prime}$, and dot notation, $\dot{x}$, are used when writing derivatives always with respect to a particular variable. They are far less cluttered visually than writing $\tfrac{dx}{dt}$. The same goes for the notation $x^{(4)}$, which is used in lieu of the visually cluttered $x^{\prime\prime\prime\prime}$ or $\tfrac{d^{4}x}{dt^{4}}$. These notations are used frequently in notationally heavy discussions in physical contexts, and the economy they provide is sometimes a virtue as it sometimes helps readability. – Dan Fox Jun 05 '18 at 07:36
  • @DanFox I am ware of that, but since this is a forum on teaching, I maintain that it is a bad choice. I have no problem with $\dot{x}$ since it doesn't suggest $x=f$. – Michael Bächtold Jun 05 '18 at 07:57
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    I disagree that it is (always) "bad notation". Mathematics is full of notation abbreviations (so-called "abuses") that greatly aid in eliminating obfuscatory cruft so that one can concentrate on the essence of the matter. A skilled writer knows how to choose optimal notation depending on the context. – Bill Dubuque Jun 05 '18 at 16:00
  • Worth mention: there is further analogous discussion of notational "abuse" in Michael's MO question on the notation $ y = y(x)$. – Bill Dubuque Jun 05 '18 at 16:06
  • @Number: thanks for highlighting that discussion. Should anyone be interested in more on this, the topic has preoccupied me for some time now. – Michael Bächtold Jun 05 '18 at 20:46
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    You can condemn the notation, but you still have to teach it. Because it is commonly used. – Peter Saveliev Nov 30 '19 at 14:16
  • @PeterSaveliev Well I teach it, but only applied to correct concept, namely to maps. If $y:\mathbb{R}\to \mathbb{R}$ then it is ok to write $y^{(n)}$. While if $y$ is a variable quantity that depends on $x$, and what you want to talk about is the $n$-th derivative of $y$ with respect to $x$ then the notation is not adequate. Do you understand that? – Michael Bächtold Nov 30 '19 at 16:07
  • @Michael Bächtold No, I don't understand. Is $\frac{dy}{dx}$ not OK? – Peter Saveliev Nov 30 '19 at 20:21
  • @PeterSaveliev It depends on what type of object $y$ is. For instance if $y$ is the map $\mathbb{R}\to\mathbb{R}$ that squares it input (so $y(x)=x^2$), then $\frac{dy}{dx}$ cannot denote the derived map since $y$ does not depend on $x$ (indeed $\frac{dy}{dx}=0$). So would write $y'$ to denote the derived map. On the other hand, if $y=x^2$, then $y$ is not a map and $\frac{dy}{dx}=2x$. In this case I would not write $y'$ for the derivative wrt $x$, since that notation does not make clear wrt which variable we derive. In essence this is about distinguishing maps from numbers. – Michael Bächtold Dec 01 '19 at 09:36