Talk:Derivative

Definition section : "df" instead of "dy" ?

After the definition is given, indications on how to read are given. They include

"dy by dx at a", or "dy over dx at a"

should it be

"df by dx at a", or "df over dx at a"

since "y" is not use in the section ? Padelsart (talk) 13:02, 1 February 2023 (UTC)

 Done Chaged dy to f, and reworded a bit. Whether or not there are people who say "dy by dx" in this context, it is a confusing thing to include right after the definition, and should be kept to the notation section. small jars tc 18:01, 1 February 2023 (UTC)

Formula for the nth derivative

I think the formula from Stegun (https://personal.math.ubc.ca/~cbm/aands/abramowitz_and_stegun.pdf, page 824) should be added here: ${\frac {d^{m}}{dx^{m}}}f(x)=m!\sum _{n=m}^{\infty }{\frac {s^{(}n)_{m}}{n!}}\Delta ^{n}f(x)$ which contains the stirling numbers of the first kind and the forward difference operator to compute nth derivatives as series. Thoughts? — Preceding unsigned comment added by Onlineuser577215 (talk • contribs) 19:35, 21 September 2023 (UTC)

Semi-protected edit request on 28 September 2023

Researchrush (talk) 23:22, 28 September 2023 (UTC)

I would like to add a brief bit about how Issac Newton called the derivative "the flowing quantity" and the notation he used.

 Not done: it's not clear what changes you want to be made. Please mention the specific changes in a "change X to Y" format and provide a reliable source if appropriate. JTP ^{(talk • contribs)} 00:02, 29 September 2023 (UTC)

Arbogast's/Euler's notation

After my reversion of the change of the heading § Euler's notation, I received the Could you please provide any evidence on the edited page that Euler used or popularized the D-notation for derivative, or that it is a "common name" (among who?) Otherwise, could you please restore my edit? Alexey Muranov (talk) 12:07, 2 October 2023 (UTC).

I am unable to provide to decide which is the most common name among "Arbogast's notation", "Euler's notation", and "D-notation", but it is certainly not "Arbogast's notation". This justifies my revert. However, there are so many things that are named after Euler's, that "Euler's notation" would need, at least, to be disambiguated. So, I'll change the heading to "D-notation", with a template {{anchor|Euler's notation}} and a note on the attribution. D.Lazard (talk) 14:05, 2 October 2023 (UTC)

@D.Lazard, I am not asking to you to decide, I am asking you to provide sources based on which you reverted my edit. Wikipedia should not invent names. A source that attributes the notation to Arbogast is currently cited in the very first sentence. You say something justifies your revert, but it is not clear what. Alexey Muranov (talk) 17:59, 2 October 2023 (UTC)

Again, per Wikipedia usage, this is not the attribution which motivates the name; it is the common usage. This is the fact that "Arbogast's notation" and "Arbogast notation" are very uncommon (less than 20 hits in Scholar Google) that justifies my revert. D.Lazard (talk) 18:34, 2 October 2023 (UTC)

Could you please provide a reference to the relevant statistics of Scholar Google which justifies the "Euler" name? Common usage by whom? How the statistics were collected? On the other hand, since it is Arbogast's notation, in the sense that he invented and popularized it, it is a sufficient reason to call it so. Alexey Muranov (talk) 20:04, 2 October 2023 (UTC)

I am actually not sure if Google Scholar statistics should be considered a source for a terminology, but it will be better than nothing. Alexey Muranov (talk) 20:07, 2 October 2023 (UTC)

I agree that Google Scholar must be considered with care. It is reliable for establishing that a terminology is uncommon. But it is not for comparing "D-notation" and "Euler's notation" because both phrases can have other meanings. Nevertheless "D-notation" seems less ambiguous. This is the reason for changing headings § Euler's notation to § D-notation. For avoiding reader's confusion, and not breaking the redirect Euler's notation (and other incoming links), I have kept an anchor, and added a sentence mentioning the attribution to Arbogast, and the alternative name of Euler's notation. D.Lazard (talk) 20:36, 2 October 2023 (UTC)

Sorry, but could you then please provide any justification that "D-notation" is a "common usage"? By whom? And if you keep a redirect for "Euler's notation", could you please explain what makes you think that this is "Euler's notation" in any sense? Alexey Muranov (talk) 04:58, 3 October 2023 (UTC)

I have absolute no opinion whether this is notation is due to Euler or not, and even whether Euler used it or not. The established fact is that many people use the phrase "Euler's notation" for naming this notation, and Wikipedia cannot change this fact, and must not try to change it. This is why, whichever is the name of the section, the redirect Euler's notation must be kept and have the corresponding section of Notation for differentiation as a target.

An title must be chosen for the section. "Arbogast's notation" is excluded because, it is very rarely used, and as such, choosing it would break the fundamental rules of Wikipedia, in particular WP:OR. So, you must chose between "Euler's notation", which is a common name for the notation, and "D-notation", which is a factual description of the notation. What is your choice? D.Lazard (talk) 08:58, 3 October 2023 (UTC)

@D.Lazard, you say """The established fact is that many people use the phrase "Euler's notation" for naming this notation""" -- please justify by providing references. According to what statistics? Do they understand what they say? "Arbogast's notation" is ok as a title, since it is indeed a notation of Arbogast. This is not orignal research, this is a known fact. At least according the literature I checked. Alexey Muranov (talk) 20:19, 7 October 2023 (UTC)

WP:CRITERIA gives 5 criteria for choosing the article title. Neither "Arbogast notation" nor "Euler's notation" satisfy the two first criteria of recognizability and naturalness. This is the reason for which I changed the heading (here and in Notation for differentiation) into "D-notation", that is both natural and easily recognizable. I have also added a sentence in the beginning of the section for attributing the notation to Arbogast, and saying that some authors (not "many") use improperly "Euler's notation". This improper use of "Euler's notation" must be mentioned since some people may search for it, at least those who learned it in Wikipedia, where it appeared in a edit of derivative of 8 August 2005‎. D.Lazard (talk) 10:55, 8 October 2023 (UTC)

GA Reassessment

Derivative

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

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Article (edit | visual edit | history) · Article talk (edit | history) · Watch • Watch article reassessment page • GAN review not found

Result: Massive effort from Dedhert.Jr and XOR'easter to bring this article up to scratch. ~~ AirshipJungleman29 (talk) 21:12, 10 January 2024 (UTC)

Significant portions, including whole sections and subsections, of this 2007 listing are missing inline citations; the article thus does not meet GA criterion 2b). If someone has access to the books of the biblography section and the requisite knowledge, this is just a matter of finding pages, to my inexpert mind. ~~ AirshipJungleman29 (talk) 00:36, 14 December 2023 (UTC)

Just as I predicted it. The article has some unsourced paragraphs during I was adding the citations. Not to mention, they are also some parts that is not understandable per criterion 1a, for example the partial derivative part. Also, the history section, to me, is somewhat not related, as it mentions the history of how the calculus was made; maybe this could be expanded or merge to any other sections, or just delete it literally? I do think there are lot of problems in this article. Pinging more users: @Jacobolus, @David Eppstein, @XOR'easter, @D.Lazard for more comments???? Dedhert.Jr (talk) 01:25, 14 December 2023 (UTC)

This article would benefit significantly from an effort to add a gentler informal or semi-formal description at the top, with less jargon, more pictures, and accessible to a wider audience. (A better history section would also be nice.) Other than that, this article seems mostly fine, and this reassessment is IMO a waste of time. Please feel free to add more specific citations if you want. All of this material is easily verifiable and discussed extensively in the cited references. –jacobolus (t) 20:17, 17 December 2023 (UTC)

Just to clarify jacobolus, are you saying that you feel this article meets the GA criteria, or that there is no need for this article to meet the GA criteria and discussing whether it should is a waste of time? ~~ AirshipJungleman29 (talk) 01:10, 18 December 2023 (UTC)

I'm saying this article seems like a fine Wikipedia article. Like most of them, it could use additional work. Arguing about whether it ticks off some boxes on a made up checklist (a poor proxy for article quality) is a total waste of time. Addressing the specific criticisms raised here isn't going to make the article substantively better, and there are many more valuable improvements that could be made (but also don't have to be; as I said it's a fine article). Knock yourself out though. –jacobolus (t) 02:12, 18 December 2023 (UTC)

The second option, then. Thanks for your quick response. ~~ AirshipJungleman29 (talk) 03:28, 18 December 2023 (UTC)

More discussion

@AirshipJungleman29, @Jacobolus: the discussion of improvement may be found in many places: see at my talk and the talk page of the article Derivative. In my talk page, me and XOR'easter discussed the improvement in the section of definition. Dedhert.Jr (talk) 03:17, 18 December 2023 (UTC)

Thanks to you and to XOR for working on the article. I hope the article will be improved, to the benefit of the readers. ~~ AirshipJungleman29 (talk) 03:28, 18 December 2023 (UTC)

Some questions from me regarding the improvement of this article:

• I'm looking for the possibility of a scenario about the notation part: should this section mention after the higher derivative? In my opinion, the notation describes many different notations of differentiation, including the high order. Dedhert.Jr (talk) 02:42, 21 December 2023 (UTC)
• Also, I wonder why its applications remain stuck at the Differential calculus#Applications. Is it possible to write them down in this article, instead? Dedhert.Jr (talk) 02:42, 21 December 2023 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Total derivative, total differential and Jacobian matrix section

The Total derivative, total differential and Jacobian matrix section is awfully wordy and detailed for an article that's supposed to be about the concept of the derivative in general. It's a bit odd that we have 10 paragraphs of text that take up about 2 screens and invoke mappings between tangent bundles, but we don't have anything on maxima and minima of single-variable functions. And within that section, there's less emphasis comparatively speaking on Jacobians than I would have expected given how much I've seen these concepts covered in courses at different stages. XOR'easter (talk) 03:43, 16 December 2023 (UTC)

If that's the case, maybe this section can be summarize in at least one, two, or three paragraphs? Also, most of them are unsourced, together with the usage of first-person pronoun. Dedhert.Jr (talk) 13:28, 17 December 2023 (UTC)

I've trimmed it back to something more reasonable. It still needs sources. XOR'easter (talk) 17:25, 20 December 2023 (UTC)

@XOR'easter I have found the source of the definition [1], but it seems likely different than the article. Hopefully this helps, and I will keep searching for more. Dedhert.Jr (talk) 01:48, 21 December 2023 (UTC)

DX - This is where I could start, where I can put my integral result with .dx

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

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I start when I could plot something in 3D (3-Dimensional). As I could do that, I apply .dx at the end for what I had found as per mathematics. On the other side it is a value we can calculate for Δy (change in y - [delta y]) instead of a rigid y. It is nothing but a change in y - Δy - f'(x).

Δy = integral_value . dx

Δy = NP_r . dx

Further I am going to find dy/dx that's the first order of derivative. Over that everything else continues like finding continuity, discontinuity, finding convergence, divergence, whether the value converges or diverges.

dy/dx = <<what's been integrated divided by dx>>

As per me, I go with dy/dx = 1 = NP₀

When I give a chance like NP₁, it's a flaw, it never going to end. as the result is n, as per me it is just not n rather Na^n-1, so the integrity collapses by muting within.

Let's have a great earth ever with our sun.—BramStoker's^t@lk 19:42, 22 December 2023 (UTC)

@BramStokers Sorry. I cannot comprehend your words and some symbols in your writings? Can you tell us in more detail? And what's with those confusing notations, such as NP₀, and so on? Dedhert.Jr (talk) 03:13, 23 December 2023 (UTC)

I was scribbling something in midnight having something in mind with my own thought process which is in obstacle always. Even when I recall / revisit myself whether it is right what I was thinking I couldn't correlate. I couldn't think stably as it's long years since my academic time. Maybe with this, it is somewhat meaningful.

y = value_of_inegral (1.dx)

It is a change. So,

Δy = dx

Thinking 1 and around it, is somewhere within my mindset.

—BramStoker's^t@lk 12:09, 23 December 2023 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Online sources

During GAR, I removed the hidden online sources while improving the article.

• Crowell, Benjamin (2017), Fundamentals of Calculus
• (Govt. of TN), TamilNadu Textbook Corporation (2006), Mathematics- vol.2 (PDF), archived from the original (PDF) on 2016-01-15, retrieved 2014-11-29
• Garrett, Paul (2004), Notes on First-Year Calculus, University of Minnesota
• Hussain, Faraz (2006), Understanding Calculus
• Mauch, Sean (2004), Unabridged Version of Sean's Applied Math Book, archived from the original on 2006-04-15
• Sloughter, Dan (2000), Difference Equations to Differential Equations
• Strang, Gilbert (1991), Calculus
• Stroyan, Keith D. (1997), A Brief Introduction to Infinitesimal Calculus
• The Notation of Differentiation, MIT, 1998, retrieved 24 October 2012
• Wikibooks, Calculus

All of these sources here are in the format CS2. Dedhert.Jr (talk) 04:48, 31 December 2023 (UTC)

Slope

So at the end, you find the value which is the area of triangle multiplied by 2, not the surface area of triangle.

Matrix(A) . Transpose Matrix (A^-1) = 1. It is the agreed one as everyone knows of it. And the matrix concepts falls on the linear algebra & part of the mathematics

I have checked in the following link, that the slope of the line can be calculated like this as mentioned in the page,- https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-linear-equations-functions/8th-slope/a/slope-formula

I had tried a little permutation of the calculation myself, concluding putting it into a two dimensional matrix.

|x1, y1|

|x2, y2|

|1, 4|

|7, 7|

x1 = 1

x2 = 7

y1 = 4

y2 = 7

Slope => y2 - y1 / x2 - x1 => 7 - 4 / 7 - 1 => 3 / 6 => 1 / 2

If it is a Matrix

|x1, y1|

|x2, y2|

Slope => (x1 x y2) - (x2 x y1) => (1 x 7) - (7 x 4) => 7 - 28 => - 21 => 21

|x, y|

|x1, y1|

At the end I am missing my mathematics that how value been profound. Number 2 is important in the mathematics also in computer field act as binary base system. Not only that, however I am not able to recall, 2 is an important number in vectors, stating the surface is 2 even if the side values are just denoted by value which is equals to 1 and it is a vector. Even √2 is important. 45° is important as it gives bijection between x and y.

And as you found a value of delta / 2 is equals to surface of the triangle value, if the z-axis is intact and the field is just 2D, and as I somehow recalls, number 2 plays vital role in vectors, I feel I can connect 21 with 1/2.

I couldn't relate 1/2 with 21 by mathematics, and as however in all higher dimensional matrix also, A.A^T = 1. So I give all my chances to the serieses like harmonic series and the other ones. Other than that slope is always profound by derivative, a division of mathematics. And in vector 2 is important. And in matrix A.A^T = 1

Let's step into 3D not by 2D any more.

—BilkTheHulk ^{Talk - "Only dead fish go with the flow."} 20:25, 17 January 2024 (UTC)

I moved this comment to a new section and archived the extremely stale 10-year-old comment it was nominally a "reply" to. I can't really make much sense of this new comment or figure out whether/how it was supposed to relate to the previous one in any way. –jacobolus (t) 23:31, 17 January 2024 (UTC)

Mathematical solutions by method / formulas has limit by connecting one another way of having mathematical explanations for it and by the ways of mathematics. Before getting into higgs, science must evolve by this way, so physics gets into a better shape. I am little wise with chemistry and it has its own limit on its applicability, so it can ignored a bit when it goes beyond its context.

I was about to say something before, but got deviated. Here it is — the way how 2 is important and √2, i is less important I feel, I mean the √-1. From derivative, or by any different chapters of mathematics, we may indulge well but we should correct our context in all sense. So, let's get into mathematical proofing and our best logical augmentation in all aspects, so we will be in a position to face consequences by all sort, the good the FLAMINGO project is the best we survive or yield to the next—AtTEnigma^t@lk 19:45, 4 February 2024 (UTC)

Sorry, but I do not understand what point you are trying to make. Please stick to concrete criticisms/suggestions about this article, and try to make your comments as clear as possible to help other editors follow along. Note that this talk page is not a general-purpose forum. –jacobolus (t) 00:48, 5 February 2024 (UTC)

Chain rule edit

@D.Lazard: Edit Special:PermanentLink/1201066147 changes constants back to all real numbers. The reversion was appropriate because it was in the context of real functions of real variables. However, I believe that it would be appropriate to have a brief acknowledgement that the chain rule is more generally applicable. I'm not sure whether it would be better inline or as a footnote. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:00, 31 January 2024 (UTC)

For applying the chain rule to a sum, a product or a fraction, one has to know the partial derivatives of a sum, a product or a division with respect to each of their arguments, and to use the chain rule for several variables. So, the proofs of these formulas using the chain rule are much more complicate (in the whole) than the direct proof. The mention of the chain rule in two variables is therefore too technical here. D.Lazard (talk) 16:33, 31 January 2024 (UTC)

Agreed: I wasn't suggesting adding an exposition on anything beyond the case of a reals function of a single real variable, just adding a brief note that it generalizes. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 19:53, 31 January 2024 (UTC)

(Aside: a link like special:diff/1201288095 will show the change in addition to the end state.) I think "real numbers" seems fine, since the context in this page is single-variable real functions. I think a more general discussion belongs at Linearity of differentiation which is where the wikilinked sum rule heading in this article points. –jacobolus (t) 16:52, 31 January 2024 (UTC)

I was suggesting only a note that there was a more general chain rule, not a more detailed discussion.

Thanks for telling me about [[special:diff/oldid]] -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 19:53, 31 January 2024 (UTC)

About

@D.Lazard: With regard to edit special:diff/, I wrote the term as used in calculus on the Real line, not the term as used on the Real line. Also, the article is not about the term as used in calculus on Euclidean spaces in general, but rather the special case of calculus on a one dimensional Euclidean space. Given that, This is about real functions, not about the real line does not explain the reason for the reversion. Would you accept {{about|the term as used in the calculus of real functions of a single real variable|derivatives on manifolds|covariant derivative|and|Lie derivative|generalizations|generalizations of the derivative|a less technical overview of the subject|differential calculus|other uses}}? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 17:34, 1 February 2024 (UTC)

As said in my last edit, hatnotes are only for disambiguation, and must not duplicate the dab page and/or the infobox. So, for this article, the only acceptable hatnote is {{Otheruse}}. D.Lazard (talk) 17:43, 1 February 2024 (UTC)