# Can $y^{(n)}$ be used as a way of representing higher order derivatives?

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

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

• It is widely used, e.g. see Lagrange's notation. – Bill Dubuque Jun 4 '18 at 18:30
• If in doubt, when you use it explain it the first time. – Gerald Edgar Jun 4 '18 at 20:06
• @Number actually it is Euler's notation – Michael Bächtold Jun 5 '18 at 5:15
• 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 5 '18 at 9:47
• @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 5 '18 at 15:51

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.

• 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 5 '18 at 7:36
• @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 5 '18 at 7:57
• 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 5 '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 5 '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 5 '18 at 20:46