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Within calculus we have many different symbols attached with differentiating (or is that deriving?) a function, probably to the dismay of the modern calculus student. So here's the setup:

As a modern researcher, you and your collaborators N, L, C, and E have just discovered a new field called "kalkulus of one variable."

Unfortunately, discord has broken out among your co-horts and they simply cannot agree on whose notation is best for your collaborative research papers and text books. As the only person who didn't come up with an original notation, they have agreed to let you arbitrate and single-handedly pick the one-and-forever notation. Whose do you pick and why?

For the following, interpret everything as a function of the appropriate variables. Also, ignore that all four of your collaborators weren't actually ever really active at the same time. The contenders (cribbing from wikipedia) have sent you their notation and a short argument:

Nutonne (Newton)

$\dot y, ~\ddot y, ~\dot{\ddot y} , \dots, \overset{n}{\dot y}$ for derivatives..err...fluxions and $\square y$ or $\boxed{y}$ for anti-derivatives...err...fluents...err..absements (yes, that symbol rendered correctly). Fundamental Theorem: $\square \dot y = y$

  • AFF: Newtonne seems to be the first of your collaborators to have developed the full theory, although he failed to email the rest of you. Claims absolute naming rights. Notation generally typesets well, unlike his nemesis's.
  • CON: There is a historical anecdote of your contemporaries floundering for a century.

Libknittz (Leibniz)

$\frac{dy}{dx}, ~\frac{d^2 y}{dx^2}, \dots, \frac{d^n y}{d x^n}$ for derivatives and $\int y ~\mathrm{d}x$ for anti-derivatives: $\frac{d}{dx} \int y ~\mathrm{d}x = y$.

  • AFF: Emphasizes the kalkulus as being about a rate of change and we always see the dependent and independent variables. $\mathrm{d}$ for difference and $\int$ from the symbol "long s" for sum. Thinks like the chain rule are particularly easy to remember.
  • CON: Screws up line spacing in typeset text. Also, is this a fraction? Can I multiply and find common denominators like I would for $\frac{a}{b}$? Better develop a theory differential forms and wedge operators...

Couchie (Cauchy)

$f', ~f'', \dots, f^{(n)}$ for derivatives and $f^{(-1)}$ for anti-derivatives: $\left(f^{(-1)} \right)^{(1)} = (f^{(-1)})' = f$.

  • AFF: nice and compact, emphasizes that we have a new function that is related to $f$
  • CON: $\sin^{-1} x$ vs $\sin^{(-1)} x$ vs $\left( \sin x \right)^{-1}$

Eyoulrrr (Euler)

$Df, ~D^2 f, \dots, D^n f$ for derivatives and $D^{-1} f$ for antiderivatives: $D (D^{-1} f) = f$.

  • AFF: Emphasizes derivatives as an operator on function spaces, looks like the "add exponents" rule for multiplication
  • CON: Pretty abstract

"A camel is a horse designed by committee"

Pick a notation for the current problem and sub-sub-field. Switch for the next problem.

  • AFF: Get to pick and choose whichever notation emphasizes the most important aspect at hand (new function / slope / operator on function spaces / ...)
  • CON: Students will hate you.

Edited to add

Please note the soft-question tag! I am personally of the opinion that we end up using whichever notation is most convenient for the work at hand (outside of Newton's notation, which is almost solely relegated to time-derivatives and physics). Furthermore, all notation is abstract and strictly symbols on a page that we have agreed define meaning (cue linguistics discussion). If it helps, think of this as a response to the hypothetical Calc 1 student asking why you notate derivatives so many different ways. Alternatively, if you had to pick one of these notations to use for the rest of your mathematical career (and the careers of all subsequent mathematicians), which would it be and why?

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  • $\begingroup$ ""Fundamental theorem: $f^{(−1)}f^{(1)}=f^{(−1)}f′=f$". Nope: that's not how you use the notation. It ought to be "$(f')^{(-1)}=(f^{-1})'=f$. $\endgroup$ – user228113 Mar 16 '17 at 9:18
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    $\begingroup$ The good thing about mathematics is that notation isn't important, it's about the meaning. As long as you clearly state what you mean by what notation, all should be fine. $\endgroup$ – vrugtehagel Mar 16 '17 at 9:43
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    $\begingroup$ @vrugtehagel: Counterpoint: a good notation for expressing the ideas of a subject really can be a major advance; e.g. if it lets you work more easily with the main ideas without being distracted by details (which are often even irrelevant details). $\endgroup$ – Hurkyl Mar 16 '17 at 10:34
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    $\begingroup$ @erfink: How is this on-topic and not opinion based? Adding the "soft-question" tag does not exclude you from the rules of the site. $\endgroup$ – Will R Mar 16 '17 at 12:47
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    $\begingroup$ Following @Hurkyl : mathoverflow.net/questions/263531/… $\endgroup$ – Ethan Bolker Mar 30 '17 at 3:33
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$f'$ is Lagrange's fault, not Cauchy's. Just kidding.

Whichever notation you end up choosing, just make sure to stay away from $\frac{dy}{dx}$. The students might start asking probing questions and the instructor might spill the beans on infinitesim... Never mind, just kidding.

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