Talk:Notation for differentiation

Initial page
There is more to be done here. but I made a start, and hope I have made all the redirects work correctly. Geometry guy 23:25, 25 March 2007 (UTC)

Missing Numerical Notation
I think this page is lacking the simplest, first day of calculus, notation where you approximate a derivative using a small delta. In other words, we need $$\frac{\Delta y}{\Delta x}$$. I was hoping to see all 3 fraction styles here $$\frac{d y}{d x}$$, $$\frac{\Delta y}{\Delta x}$$, and $$\frac{\partial y}{\partial x}$$, with an explanation for when to use each. The rules are that the delta style (the missing one) is the numerical approach where you then take the limit as the delta goes to zero. The non-script $$d$$ is for a single-variable function, and the script $$\partial$$ is for a multi-variable function. I think the delta-style (whatever you call it) is necessary to show since (1) it's a valid means to express a derivative, (2) it's a very common notation used when teach derivatives, and (3) it'll complete the picture painted on this page. Hey.Dr.Nick (talk) 15:58, 6 January 2021 (UTC)

Future work

 * Partial derivatives should be introduced prior to using them under vector calculus. Right now, they aren't "defined" until the end of the article.
 * There is no mention of mixed partials currently. This is important because I came to this page trying to see how to "read" the order of the derivatives.  For example, does $$\frac{\partial^2f}{\partial x \partial y} = f_{xy}$$ mean "the derivative of f with respect to x then with respect to y" or vice versa? Kdmckale (talk) 17:35, 10 August 2010 (UTC)
 * That's the same reason I was reading the article. The answer can be found in the article Partial derivative: $$\frac{\partial^2 f}{\partial y \, \partial x} = \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) = f_{xy} = \partial_{yx} f$$. Note that in many cases it doesn't matter, see Symmetry of second derivatives. Paul Breeuwsma (talk) 20:01, 7 December 2010 (UTC)

It would be nice to see examples provided for laypersons and even those versed in the subject to see some concrete examples.

Suppose instead of showing how to write the derivative at a particular point using dy/dx (a) but showing several examples for different functions.

When one wants to write the first derivative of the gamma function for a particular point, should the "|" go between the function and the dy/dx...can the "y" be left out, and should the dx be in the beginning or after the written out function?

I myself am wanting to write the derivative of a specific form of a function at a particular value and so this leads to notational problems that I believe are not addressed in any consistency.

For AGM(1,sqrt(1-x^2)) how would one write the derivative of this function at a particular value? I have done so in several ways:

(d/dx)AGM(1,sqrt(1-(x=a)^2), (d/dx)AGM(1,sqrt(1-x^2))|x=a, (d/dx)(x=a)AGM(1,sqrt(1-x^2))

So, let's see plenty of examples from the experts. Numbertruth (talk) 21:43, 1 January 2013 (UTC)


 * Funny - just switched the sections around before reading your post and writing that I did so on this talk page... M&and;Ŝc2ħεИτlk 08:19, 26 March 2013 (UTC)

p,q,r,s,t Notation
Some books on PDE use the following notation in first and second order PDE: $$p=z_x$$, $$q=z_y$$, $$r=z_{xx}$$, $$s=z_{xy}$$, $$t=z_{yy}$$. I have no idea where this notation originated, or how common it is. Hadaso (talk) 07:17, 25 October 2013 (UTC)

uncommon notation
The following is without reference and is not commonly used:
 * the use the nabla operator using subscripts to indicate the variables with respect to which the function is being derived:
 * $$   \nabla_t f = \frac{\partial f}{\partial t}. $$

I am going to take it away from the article. Bo Jacoby (talk) 11:14, 23 September 2014 (UTC).

"Euler's" notation (D operator) mistaken?
The only reference to notations attributes this to Arbogast. See http://jeff560.tripod.com/calculus.html. And see our own, https://en.wikipedia.org/wiki/Differential_operator#History. — Preceding unsigned comment added by Brucer42 (talk • contribs) 18:14, 8 October 2014 (UTC)
 * Cajori's History of Mathematical Notation also attributes the differential operator to Arbogast. However, as these things go, if Euler popularized the notation – as he did for many others – it is likely that the notation would get named after him rather than the inventor of the notation. Bill Cherowitzo (talk) 20:46, 8 October 2014 (UTC)


 * But in the article "The History of Notations of the Calculus" by Florian Cajori, it says that Arbogast made this notation public in 1800, seventeen years after the death of Leonhard Euler (p. 7). So it seems unlikely that Euler would have popularized it before that... 31.211.214.225 (talk) 17:54, 6 January 2015 (UTC)Emelie

I'm going to agree with the comments by Brucer42 and Emelie here. Euler was much more focused on differentials themselves, rather than on derivatives, so the notation would have made little sense for him. However, according to Cajori, the D operator *was* used by Bernoulli (a friend of Euler), however, this usage was different. Bernoulli's D operator was for *finite* differences. Given that Cajori, when he mentions Euler, only mentions differentials, and lists the D operator to Arbogast, Pierce, and Peano, I think that this should not be called Euler's notation. In fact, I think that this may actually be fostering some confusion on the Internet, as some people are referring to this as Euler notation, probably based on this article. johnnyb (talk) 14:23, 23 January 2018 (UTC)


 * Agreed. I checked Euler's Mechanica, sive, Motus scientia analytice exposita v1 1736 and Euler's Institutiones calculi differentialis, 1755 that could be found on https://archive.org. There is no "D operator". There are dy, ddy, d3y and v=ds/dt
 * Book Institutiones calculi differentialis, 1755 explicitly says on page 126 right: we use differentials defined by LEIBNIZIO. SMZinovyev (talk) 03:23, 10 July 2023 (UTC)

Notation for integration
I have no idea why a section on notation for integration has been added. We already have an article for that: Integral symbol. The section talk:Notation for differentiation should be moved over there. 'M'&and;Ŝc2ħεИτlk 15:50, 5 February 2016 (UTC)

Rewording done and more subsections have been added. It's notation for the antidifferentiation (or indefinite integration), the opposite operation for the differentiation. Integral symbol is about ∫ character. Newton's notation redirects here. There isn't a Notation for antidifferentiation article. I think this is the right article where they can be compared.79.150.132.55 (talk) 18:44, 7 February 2016 (UTC)

Does anyone know of source for the Liebniz notation for a double antiderivative? Specifically the dz squared part at the end. — Preceding unsigned comment added by 64.235.154.42 (talk) 15:55, 11 September 2018 (UTC)

Assessment comment
Substituted at 02:24, 5 May 2016 (UTC)

International Standards
The notation of differentials, derivatives etc. is also standardized in ISO 80000-2 Quantities and units — Part 2: Mathematical signs and symbols to be used in the natural sciences and technology (this title is intended to be shortened to read “Mathematics” in a future edition.) and other related international and national standards. This reference and the conforming notation and typography should be included in the article. – Emeldir (talk) 13:05, 16 January 2017 (UTC)


 * In the past, ISO's mathematical typography has always struck me as uninformed, amateurish, and ugly. Moreover, their prescriptions are never used within pure mathematics and are not consistently used within applied mathematics. While it may be necessary for this article to mention ISO, I feel like it would be undue weight to include more than a brief paragraph. Ozob (talk) 18:38, 16 January 2017 (UTC)


 * I personally prefer the upright d for derivatives yet I see we have used the italicized version here for this article. Most people in my department (Applied Mathematics) use the upright d, as well as the pure maths department upstairs. Is there a specific reason why the italic d was used for this article. Kendall Born (talk) 07:38, 3 December 2019 (UTC)


 * I find that italic d is used more by pure mathematicians while roman d is more often used by physicists, engineers, and applied mathematicians. (There are exceptions in either direction, of course.) Since this article is about a mathematical topic, I think it makes more sense to use the stye from pure mathematics. That said, the roman style deserves a mention, which I've added. Nloveladyallen (talk) 05:40, 4 December 2019 (UTC)

Boxes on the right-hand side
I'm not sure why we need the boxes on the right-hand side illustrating the use of each notation. Each notation is already demonstrated in its respective section, so the boxes are redundant, unless your attention span is so poor that you can't read the article text, and need everything important in VERY LARGE FONT.

If we are going to keep them, they should at least be set in TeX. The HTML versions are extremely ugly, at least on my machine. Nloveladyallen (talk) 05:56, 2 December 2019 (UTC)

Lagrange's Notations is actually due to Euler
See: https://hsm.stackexchange.com/a/7409/3462

This should be reflected in the article. — Preceding unsigned comment added by Mbachtold (talk • contribs) 10:54, 20 December 2020 (UTC)
 * Nice! It took me a couple of minutes to fix the article, but how should we call the notation then? Ain92 (talk) 12:13, 19 January 2021 (UTC)

Numerical suffixes
What seems to be missing from this article is the convention for using numbers instead of names of variables for partial derivatives, e.g. if $$f$$ is a function of $$x$$ and $$y$$, writing $$\partial_2 f$$ or $$D_2 f$$ instead of $$\partial_y f$$ or $$D_y f$$ or $$\partial f / \partial y$$ (i.e. differentiation with respect to the second variable).  Dr Greg  talk 22:27, 28 July 2022 (UTC)


 * I’ve never personally seen this notation before. Although I can see it being a bit confusing: given that the order and choice of variables for a multivariable function aren’t necessarily determined, other than by tradition and convention, this notation is a bit ambiguous and could lead to different results for different choices of variables. I think this notation creates a mathematical dependency on specific variable choices and their ordering that doesn’t exist in most, if not all, of mathematics. Do you have multiple references where this notation is used? WikiMathematician (talk) 19:48, 31 July 2022 (UTC)
 * I completely forgot to mention my second reason for potentially confusing notation: higher-order differentiation. This is less likely to be as significant an issue as the other reason, given the standard position demarcating this differs, but a potential issue nevertheless. WikiMathematician (talk) 19:56, 31 July 2022 (UTC)
 * One last note: I can see a very specific use-case for this numerical notation, where a single variable with subscripts is used. In that case, it’s more of an issue of convenience and probably used as an alias rather than a standard convention, e.g. an excerpt that reads something like this:
 * “For convenience, we will notate $$\partial_{x_n} f$$ as $$\partial_n f$$…”
 * WikiMathematician (talk) 20:09, 31 July 2022 (UTC)


 * The undergraduate maths text book Apostol introduces the notations $$D_k f$$ or $$f_k$$ as alternatives to $$\partial f / \partial x_k$$ (for $$k = 1, 2, ... N$$) for a function of $$N$$ real variables, and prefers to use $$D_k f$$. This is at the point where partial differentiation is defined for the first time. Although you say that the order of variables doesn't exist in most cases, in terms of rigorous pure maths it's really the other way round, the variables have an order but don't have names. You can choose to write a function as $$f(u, v) = u + v^2$$ but that's exactly the same function as $$f(v, u) = v + u^2$$ or $$f(\alpha, x) = \alpha + x^2$$. And in all three cases it would correct to say $$D_2 f(p, q) = 2q$$. For what it's worth Apostol uses $$D_{r,k} f = D_r(D_k f)$$ for higher order derivatives.


 * Perhaps the best way to express this in the article, without getting bogged down in pedantic detail, is just to say that $$D_k f$$ is used by some authors instead of $$\partial f / \partial x_k$$, with Apostol as a reference, and make no further comment. (This was how I was taught in university pure maths lectures, but the applied maths lectures used the more familiar $$\partial f / \partial y$$ notation.)


 * The notation $$\partial_\alpha f$$ is common in tensor algebra and more specifically Ricci calculus. You can see it being used, for example, in Covariant derivative (where you can see yet another notation $$f_{;\alpha}$$). Probably in several other articles too, but it may take some time to find them. However, on reflection, this is quite an advanced application so probably $$\partial_\alpha f$$ needn't be mentioned in this article. I still think $$D_k f$$ is worth a brief mention, though. --  Dr Greg  talk 19:23, 1 August 2022 (UTC)

Was Leibniz bad at algebra?
The rationale given for higher-order derivatives is that it's from "formal manipulations of symbols", i.e.


 * $$\frac{d\left(\frac{dy}{dx}\right)}{dx} = \left(\frac{d}{dx}\right)^2y = \frac{d^2y}{dx^2}.$$

But as any junior high school algebra student knows, it should be


 * $$\frac{d\left(\frac{dy}{dx}\right)}{dx} = \left(\frac{d}{dx}\right)^2y = \frac{d^2y}{(dx)^2}.$$

What was Leibniz thinking? jej1997 (talk) 07:59, 8 March 2024 (UTC)


 * In this context, $d$ is not a variable and must be seen as the symbol of a function. Thus $$dx$$ must be interpreted as $$d(x),$$ and both  $$dx^2$$ and $$(dx)^2$$ must be interpreted as $$d(x)^2=(d(x))^2.$$ At Leibniz time, the functional notation was not yet fixed, and the parentheses around the argument of the argument of a function were not systematic. For example, one writes still $$\sin x$$ instead of $$\sin(x).$$ D.Lazard (talk) 09:34, 8 March 2024 (UTC)