Talk:Derivative

GA Review

GA review (see here for criteria)

1. It is reasonably well written.

   a (prose): b (MoS):

2. It is factually accurate and verifiable.

   a (references): b (citations to reliable sources): c (OR):

3. It is broad in its coverage.

   a (major aspects): b (focused):

4. It follows the neutral point of view policy.

   a (fair representation): b (all significant views):

5. It is stable.

   

6. It contains images, where possible, to illustrate the topic.

   a (tagged and captioned): b lack of images (does not in itself exclude GA): c (non-free images have fair use rationales):

7. Overall:

   a Pass/Fail:

As 2:b is disputed, I'll declare it as void, and approve this article. _→AzaToth 13:01, 10 October 2006 (UTC)

Additional remarks

This article has great impact as it covers a topic that is not too esoteric and is familiar to most encyclopaedia readers. As another GA reviewer, I would like to add some suggestions on improving this article. Some of these suggestions are simply so that the article conforms to the mathematics Manual of Style and the Wikipedia-generic Manual of Style guidelines. These suggestions should be implemented as soon as possible so that the article is representative of a Good Article.

• The current lead section is broken up into short two-sentence paragraphs with a bulleted list included within it. Following the guidelines of the MoS on the lead section, this section should be rewritten into two or three paragraphs without the bulleted list. It should provide an overview of the main points and should be capable of standing alone. Self-referential statements such as, “The remainder of this article discusses only the simplest case (real-valued functions of real numbers)” should be avoided. The main text covers the simplest case and a final section adds information for generalisations, so such statements are not needed.
• According to the mathematics MoS, there should be some introductory paragraphs, which describes the subject in general terms. This is particularly important for a basic mathematics article. Although this article is supposed to be a break-out article from the main Calculus article, an introduction section would make the article complete in itself which is important if the reader came across the article from another path. It has also been suggested to connect this article with the History of calculus article.
• Also as noted in the mathematics Manual of Style, the use of the first person (“we”) should be avoided.
• In the Physics section, the first clause “Arguably the most important application” is an example of WP:WEASEL words. If it is the most important, then just say so, otherwise please provide a citation where someone states that it is an arguable claim. Another example is “it is usually rather easy to get a rough idea of…”. This should be reworded or dropped.
• As this article covers an uncontroversial topic, it would help to cite to one or two authoritative sources at the start of the article in order to conform to verifiability criteria. The Physics Wikiproject has a proposal on how to deal with this kind of citation. Please consider adding this cite to the sources you have listed under References.

Other comments:

• In Leibniz's notation section, the last sentence is a parenthetical element. Is there a reason why?
• The Physics section has a good examples. Could other examples be provided from other fields? If necessary, the physics example could be shortened to add other examples.

I hope that you find these suggestions helpful in improving your article on the way toward Featured Article status. RelHistBuff 11:51, 11 October 2006 (UTC)

Recent revisions have addressed these helpful suggestions in the following way.

• The new lead section now comprises four short paragraphs and stands alone .
• The idea of differentiation is introduced in the first section before a formal definition is given.
• The History of Differentiation has been expanded using material from the History of calculus and other sources.
• The first person pronoun has (almost?) entirely been eliminated.
• Weasel words have been eliminated.
• The parenthetical element at the end of Leibniz's notation has been replaced by a footnote.

Authoritative sources have not yet been chosen for the start of the article and examples from other fields have not yet been provided. I believe that the latter cannot be done properly until a separate article on applications of derivatives has been written. However, the lead does now at least refer to the fundamental application of differential equations, and mentions applications in economics. Further comments on this article are always welcome. Geometry guy 18:23, 31 March 2007 (UTC)

rate of change - Änderungsrate (de)

A discussion about the lemma Änderungsrate in the german wikipedia lead to the conclusion, that (momentary) "rate of change" (Momentane Änderungsrate) in applied mathematics (e. g. physics) is not exactly the same as analytic derrivative. Therefor we encounter some problems in interlinking our article on Änderungsrate with the english wikipedia (interwiki). Could someone familiar with the english terminology help here? --KleinKlio 17:22, 11 October 2006 (UTC)

English "derivative" = German "Ableitung". However, the German wikipedia does not have a separate article about the mathematical meaning of that term, so I changed the interwiki to point to de:Differentialrechnung which seems the best match. -- Jitse Niesen (talk) 02:57, 12 October 2006 (UTC)

I'm sorry, I misinterpreted your question. You're looking for the proper interwiki link for de:Änderungsrate. Unfortunately, I can't find a good target in the English Wikipedia. (Note for other editors: I think that the central point at the discussion on the German WP is that "(instantaneous) rate of change" can only be used if the function depends on time, while "derivative" is more generally applicable). -- Jitse Niesen (talk) 03:14, 12 October 2006 (UTC)

Thank You for Your help so far, Jitse Niesen. Your decision to interwiki "derivative"- "Differentialrechnung" seems quite right to me. There's a lot discussion about how large (in terms of content, not words) an article should grow, but for the moment there is no "Ableitung". The article de: Änderungsrate does link to "Differentialrechnung" too, when it comes to mathematical details.

The german "Änderungsrate" is like the english "rate of change" used in a metaphorical way synonymous to derivative=Ableitung, but since in experimental physics you can sometimes measure a (formal) derivative in a independent way (for example electric current independently from electric charge), we desided to develop an extra arcticle for this. This was the main justification for doing so: Änderungsrate is (for Germans) not really a special case of derivative. But perhaps are all those german Gedankenexperimente? :-) --KleinKlio 17:01, 12 October 2006 (UTC)

By the way (I'm not a crack in English language, pardon me): Should the disambiguition sentence in the intro not say "For other meanings of this word, see ..."? --KleinKlio 17:43, 12 October 2006 (UTC)

represent

Can anybody explain me what "social science apps often use derivative + or - sign to 'represent' empirical-theory relation" or "derivatives can be used to represent many properties of a function" is supposed to mean? English is not my native language, but I know it pretty well and "represent" seems like the wrong word here. -- Jitse Niesen (talk) 08:33, 25 November 2006 (UTC)

Well, I hope that I can (although you might not be pleased with my answers;):

"derivatives can be used to represent many properties of a function"

I take it that the only term you find puzzling there is 'represent'. But in the article there [was] a [Wiktionary] link to 'represent', which if you mouse-click will take you to definitions of that term. The definition intended is #6. (One way of flagging the intended definition in the article would be include '[#6]' or '[def. 6]' after '(or represent)'.) (This was referenced in the Edit summary, which you might not have seen.)

"social science applications often use a + or - sign of a derivative to 'represent' a hypothesized relation" [slightly expanded and rephrased]

Take my field, economics (please). There it is common and standard to place a +, -, or ? sign over each argument of a function. The signs refer to the (so-called qualitative) relationships implied or hypothesized by the respective partial derivatives (see Samuelson's Foundations of Economic Analysis for a similar use). Equivalently, partial derivatives may be referenced directly, as for example by this shorthand:

δGDP/δM > 0, δGDP/δGovSpending > 0, . . .

Or, in notation for a function of only one independent variable:

dGDP/dM > 0.

These are examples of 'representing'. I appreciated your allowing me to respond, rather than simply reverting (although reverting, preferably with an Edit summary, might be quite appropriate, constructive, or otherwise helpful in many instances). Thomasmeeks 14:05, 25 November 2006 (UTC) Edited by Thomasmeeks 00:11, 26 November 2006 (UTC)

I don't quite see how this is "representing"; I may have to get a copy of Samuelson. The partial derivatives are what they are, and information can be inferred from the signs of them. I see this as a use of the derivative, not that the derivative is doing some "representing". Perhaps this is simply a matter of perspective. In any case, since this is the lead of the article, I think simplicity (without misleading) is paramount, and since "representing" is not, apparently, a simple aspect of derivative, perhaps it could be included elsewhere (say, a section somewhere on applications of derivatives in economics). Or, perhaps it could be split into another sentence so that we have one on the derivative's use to determine properties of a function, and another one on "representing". Note that the first sentence of this 2nd lead paragraph implies that the paragraph is about real-valued, single variable functions, for which partial derivatives are not applicable; so perhaps you could say something about representing in the 3rd paragraph or 4th paragraph? Doctormatt 22:46, 25 November 2006 (UTC)

Thanks for your comment. On the last, you might be right about breaking them out. I'm more focussed on the immediate question right now. I'm afraid that on that, the point I was trying to make above did not come out as plain as it should have. Let me try to respond more directly to your point. Wiktionary def. #6 of 'recognize' at represent' is:

To serve as a sign or symbol of; as, mathematical symbols represent quantities or relations; words represent ideas or things.

That seems to correspond closely to the above usage, closer than 'use' & close enough to everyday use to be helpful (possibly with a link and cite of def. 6). Thomasmeeks 03:12, 26 November 2006 (UTC)

2nd sentence of 2nd paragraph edited

2nd paragraph, 2nd sentence: Earlier edit: Derivatives can be used to determine (or represent) many properties of a function, including whether the function is increasing or decreasing on an interval and whether it has a maximum or minimum at a point.

Current edit: Derivatives can be used to characterize many properties of a function, including

• whether and at what rate the function is increasing or decreasing through a value of the function
• whether and where the function has maximum or minimum values.

Advantages: (1) It is more specific about the uses of derivatives without being complicated. (2) It removes the parenthetical suggestion of the earlier edit that 'representing' is secondary. (3) The wording of "the function is increasing or decreasing on an interval" is avoided. While this is doubtless good shorthand, someone unfamiliar with the usage might well be puzzled (unless he or she studies the increasing or decreasing links). Thomasmeeks 22:28, 27 November 2006 (UTC)

Style of introduction definition

I changed the definition of a derivative in the first sentence from:

In mathematics, a derivative is defined as the instantaneous rate of change of a function. The process of finding the derivative is called differentiation.

to:

In mathematics, a derivative is the instantaneous rate of change of a function, which is to say the rate of change at a particular point in time (or some other variable) and not the average rate of change over a period. Finding a derivative is called differentiation.

I believe that the original definition, while to-the-point, does not make sense to the person who does not already understand calculus. For example, what is an "instantaneous" rate of change? No one talks about the needle of a speedometer pointing to the "instantaneous" speed of a car. It's an obvious jargon term. My revision also removes the unnecessary "The process of" which is redundant.

Doctormatt reverted my edits. I hope we can discuss this ... otherwise I certainly feel that my version, while not perfect, is at least better than what is here.

Great - let's discuss. My feeling is that the very first sentence should be unequivocal, even if it uses a potentially confusing term (instantaneous). Your version is too confusing, with various clauses that, while well intentioned, don't make things any clearer. Why compare instantaneous rate of change to "rate of change over a period"? You claim that original first sentence does not make sense to the person who does not already understand calculus, but who other than someone who has studied calculus has used the term "rate of change over a period", or, for that matter, "rate of change of a function"?

I really think we can do better than to have a "which is to say" phrase in the very first sentence of such a fundamental mathematical concept.

(Truth be told, I don't like this "definition" at all. The derivative is not defined as the instantaneous rate of change of a function at all. This is merely one interpretation of the derivative (another is the slope of a tangent line). But I assume that there is some kind of general agreement (and the history of the page shows much variation) here that this is the way folks want the definition to be stated. So I'm working from that.)

Your second sentence is, I believe, not grammatically correct. I think yours should be "Finding a derivative is differentiating", or perhaps "Finding a derivative is called differentiating". Differentiation is the process of finding the derivative, as stated in the original version. By the way, I don't think there is any reason for this to be the second sentence in the article. I suggest moving it further down the page. Cheers, (and don't forget to sign your comments with four tildes) Doctormatt 04:05, 8 December 2006 (UTC)

Hi Doctormatt, perhaps we could say that "a derivative is the rate of change of a quantity (e.g., function) at a certain time." That seems fairly unequivocal and much more accessible than what is currently there. I'm not so much a proponent of my own version so much as an *opponent* of the use of the meaningless (to the non-mathematical) word "instantaneous."

I don't really mind the differentiation thing being right after the initial sentence, because people might have clicked on the derivative page after going to differentiation (a disambig page). It's certainly grammatical to say "Finding a derivative is called differentiation"; see gerund. However, now that I think about it, maybe we should just say "Differentiation is the process of finding a derivative." (We put the sentence in the active voice.)

Your thoughts?

Best,

128.36.70.147 06:32, 8 December 2006 (UTC)

Well, I think perhaps this is better, but I don't like the "e.g." bit. Again, I think the first sentence should be as definitive as possible. Since you went ahead and changed the article, I'll go ahead and change it, too, so it doesn't have this "for example" kind of thing. Cheers, Doctormatt 05:46, 10 December 2006 (UTC)

Okay, I didn't change it. Frankly, I think the new sentence with the "e.g." just sounds terrible. Perhaps the first sentence could talk about quantities OR functions, but not both? Also, "at a certain time" suggests the question: "what time?" I think "at an instant" or bringing back "instantaneous" would be more clear. Cheers, Doctormatt 06:00, 10 December 2006 (UTC)

Hi Doctormatt, I went ahead and changed it because you hadn't written back. Thanks for your comments. What if we changed the first paragraph to:

"In mathematics, a derivative is the rate of change of a quantity. A derivative is instantaneous, or calculated at a specific instant rather than as an average over time. The process of finding a derivative is called differentiation. The reverse process is integration. The two processes are the central concepts of calculus and are related via the fundamental theorem of calculus."

My main complaint was the word "instantaneous," which I think the common man has a hard time understanding. When something is instantaneous, it is fast. There other definition, "pertaining to an instant," is rather obscure.

In the revised version, we clean up instantaneous, but also explain what it means in the next sentence. Moreover, we keep the first sentence decisive.

Hope to hear your thoughts soon. 128.36.70.147 07:05, 12 December 2006 (UTC)

Okay, this is better. Can the second sentence be written to avoid the "or"? Would "A derivative is instantaneous: calculated at a specific instant..." be acceptable? The "or" might lead to confusion, since it is not clear that the clause following it is a definition, as opposed to just being an alternative. Cheers, Doctormatt 17:06, 12 December 2006 (UTC)

Done. 128.36.70.147 23:35, 13 December 2006 (UTC)

Thanks to mathematicians, consolation to non-mathematicians, & a plea

The "thanks" is for your interest in Wiki math articles as readers and editors. What a compliment to non-mathematicians (like me) whose fumbling efforts are corrected or improved (or removed where appropriate), by you, often with an Edit summary or Talk Page reference. This is in the best spirit of critical rationalism. The principle of charity is thereby exhibited (where there is any truth worth salvaging, of course).

For bruised non-mathematicians (sometimes including me), take consolation. Corrections or deletions by mathematicians are rarely personal. The mathematicians are just trying to improve the article. Rather, indifference ("Why bother with this mess?") may be the highest form of contempt.

The "plea" referenced above is to mathematicians, but it applies to any professional field, including my own (economics): Your writing with the lay public in mind, especially in the lead, is sooo appreciated. Exposition or usage that is common, even preferred, in the field may come across as "jargon" to interested outsiders. If you Edit for the interested, intelligent layperson, that should be sufficient, provided that the exposition is relatively self-contained (rather than being circular) or that it has transparent links. If on the other hand one term or sentence is explained with another unexplained (or more obscure) term or sentence or with an obscure link, the reader may give up, even though only a little more thought and editing might render the term or sentence clearer. Yes, terseness is a great virtue but not at the cost of clarity, especially in an encyclopedia. Given the beauty, clarity, and logic that mathematician rightly take pride in, this might be "preaching to the choir." Still, y'all know what I'm talking about (Find Lefschetz). Thanks again. Thomasmeeks 15:50, 25 November 2006 (UTC)

Explicative Numerical THEORY

Dosent exactly has to do with derivatives, but EN Theory should revolutionize Diferential Calculus if proved.

If T=Lim of x>1, then. T+1 (c) = Lim >I * dy/dx (T * e) —The preceding unsigned comment was added by Hanek45 (talk • contribs) 19:28, 11 December 2006 (UTC).

cleanup needed

Someone needs to clean up the references. Spivak has written many books, I have no idea which of his books the reference is to. Also, the first reference needs an ISBN. Rick Norwood 14:50, 31 January 2007 (UTC)

Go ahead. :) Oleg Alexandrov (talk) 16:20, 31 January 2007 (UTC)

This article has to do too much work

Reading through the talk page, I see that there have been many issues concerning this article, which seem to have been brushed under the carpet since it acquired GA status. I don't believe these issues have been adequately resolved, mainly because this article does not or should not exist in isolation. However, at present, it is not only the effective main article for the differential calculus category, but also one of the few accessible and reasonably polished articles in the entire category. Because of this, it has to be everyman's article, and has too much work to do. Many of the problems it has (as a consequence) have been raised already, but I think it is worth repeating three of them.

• The article is too long. It covers in detail things like "notations for differentiation" and "rules for finding the derivarive" which really belong in separate articles. Furthermore, it discusses separately three applications, to physics, critical points, and graph sketching, whereas a general article would be much better off with an applications section and links to articles with further details.
• The article is too elementary throughout. A wikipedia article should progress. I agree this article needs to be really really accessible, but at present, it is more elementary than the differentiation section of the calculus article which it cites for a non-technical overview! (The latter article even describes the derivative as an operator! In part, of course, this is a problem with the calculus article. Can we help using material from here?)
• The article restricts attention to the derivative in one variable. I know this has been discussed already, but believe the discussion has been coloured by the amount of work this article has to do. I agree that there is scope for an article entirely on the derivative in one variable, but it should not be called "Derivative", at least, not without disambiguation. At the moment the differential calculus articles (and some of the discussions) convey the mood that one either works in one dimension or in Banach spaces. This is ridiculous. The mainstream of calculus is finite dimensional but not necessarily one dimensional (e.g. what could be a more central and motivating concept in analysis than a partial differential equation?). A fundamental article on the derivative needs to make contact with partial derivatives and the derivative as a linear map or matrix.

I saw one light in the history of this article when Dmharvey pointed out that

The common thread is that the derivative at a point serves as a linear approximation of the function at that point. I think it would be great if something like that could appear in the intro.

It would indeed. It is a pity that this opportunity was missed, as this is an idea that should inform the entire differential calculus category (with exceptions to this idea, such as the Gateaux derivative, duly noted).

This light does not shine at the moment. The multivariable calculus article is almost content-free. Total derivative cannot make up its mind what it is about, and has several vague ideas with no connections between them. The derivative as a linear map is spread all over the place, from bizarre locations such as differential (infinitesimal) to technical and specialized ones concerned with Banach spaces or manifolds.

Despite my concerns about this article, I am mainly posting here because it is one of the best articles in differential calculus, and has many editors. So please comment, whether or not you agree with the above, on how we might hammer the differential calculus category into better shape. Geometry guy 23:42, 23 February 2007 (UTC)

I've now "improved" (I hope) total derivative and differential (infinitesimal), but there is still no satisfactory "homepage" for the derivative in more than one variable. My current thinking is to create a new page at derivative (differential) if I have the energy, unless someone suggests an alternative way forward. Geometry guy 22:57, 28 February 2007 (UTC)

Recent edits

There is an anonymous user replacing italic d with roman d in many articles (see Talk:Integral). I believe these edits are in good faith (although the user's edits in general have been variable and suggest he/she may be a teenager - by which I mean no offense). However I do support them. I think the upright d works particularly well in wikipedia, because of the mixture of wiki-text and math. Geometry guy 00:06, 23 March 2007 (UTC)

Well, I went and reverted them all, sorry. We had a discussion a while ago about that, and people were in favor of italic d (which I think is the standard in US, the upright d seems to be preferred in Europe). Perhaps we can have another discussion at Wikipedia talk:WikiProject Mathematics on the issue. Oleg Alexandrov (talk) 03:55, 23 March 2007 (UTC)

Shame on you Oleg, for referring to prior history to justify your revert without providing a link to back up your assertion! :-) The only discussion I know of is at Wikipedia talk:WikiProject Mathematics/Archive 4#straight or italic d? and I see no consensus there at all. The voice of reason, in my view, is Toby Bartels, who says

I oppose any sort of policy decision for all articles; we should follow the usual rule of tolerance for variation that applies to US/UK spelling differences.

This is more subtle than a US/UK thing: it is a matter of taste with users on both sides of the atlantic, with both opinions held on both sides. There are also mathematical divergences, for instance depending on whether dx (isn't that pretty?) is viewed as a single symbol or the differential of a function. And, as the anonymous editor writes in the edit summary at Integral

Upright d for derivative notation so as not to confuse with d*x notation

For all these reasons, I generally don't like it when people (from the most humble anonymous visitor to the most esteemed administrator) go through articles uniformizing stylistic differences like this. As long as an article is internally consistent, I am okay with variations: I would even support using upright d in wiki-text and italic d in display math. However, in this case the anonymous users edit summary is somewhat better than Oleg's

rv \rm d. Ugly, against Wikipedia conventions.

Hmmm... shades of POV? :-) Anyway, I'll commit the same crime and revert here and at differential form, but leave Oleg's revert intact at Integral. Let's see what happens ;-) Geometry guy 10:41, 23 March 2007 (UTC)

However, the problem is that the italic d notation has been there for a while. It was not me who was going through articles doing mass changes, I just revered back to the notation which has been used in these articles for years. A discussion on this is in order. Oleg Alexandrov (talk) 14:44, 23 March 2007 (UTC)

Yes, you are completely right. I've now found quite a lot of discussions on this issue, and I would guess you are fed up with it cropping up again and again, and constantly having to revert mass changes! To make up for causing trouble, I'll try to put together a list of links to the previous discussions. On the other hand, although, as I said, I generally don't like mass changes to uniformize, it is not so bad if they happen randomly once in a while, as it adds to the diversity and liveliness of wikipedia (with benefits e.g. for NPOV). In this case I also felt a certain sympathy and wish not to bite the newbie.

As for further discussion, from my point of view, the most interesting point concerns the exterior derivative (cf. differential form in the above revert), since I would quite like to make that particular d upright to highlight its status as an operator. For derivatives and infinitesimals, the notation has many interpretations, and we will never reach consensus: I certainly won't revert again if the d's here turn back to italics.

Regarding this article, I think there are much more serious issues to tackle, and I raised them in the previous section. I'm sorry that no one has added further comments to these, so I should probably be bold to rekindle some interest :-) Geometry guy 15:15, 23 March 2007 (UTC)

I think this is a color of the bikeshed type discussion, which we could argue about for eons. Consistency within one article is good consistency across all of the mathematics articles is a grand waste of time. --Salix alba (talk) 15:57, 23 March 2007 (UTC)

I agree striving for consistency across articles is not really worth it.

I reverted the upright d notation per discussion at Wikipedia talk:WikiProject Mathematics where there was good agreement that notation change is likely to cause more trouble than what it's worth.

As far as differential forms are concerned, I understand that some people may prefer using an upright d notation to distinguish the exterior derivative operator from the form it is applied to. I will not try to change notation there one way or another (up to Geometry guy, that is). Oleg Alexandrov (talk) 02:37, 24 March 2007 (UTC)

Very sensible compromises have been made here I think. I'm happy to keep differential form as it is and will not change exterior derivative until I have the time and energy to make substantial improvements. Meanwhile, what about substantial improvements to this article? ... (Comment moved to new section). Geometry guy 21:05, 24 March 2007 (UTC)

Improvements needed? If so, what?

(continued) ... Meanwhile, what about substantial improvements to this article? I am tempted to ship out some of the material to subarticle stubs to focus our attention, but I'm quite busy now and won't do it just yet, so please comment or (better) edit! Geometry guy 21:05, 24 March 2007 (UTC)

Overall I like the derivative article. What is in here that you want to "ship to subarticles"? Oleg Alexandrov (talk) 03:26, 25 March 2007 (UTC)

It's a bit ironic, after all this talk about notation, but I think the "Notation" section in the articles is too long and should be condensed. So one possibility would be notation for derivatives; I can envisage this growing in a proper article with tracking the histories of all these notations (well, that's perhaps a bit optimistic, but hope springs eternal). -- Jitse Niesen (talk) 12:05, 25 March 2007 (UTC)

I agree with this: the notation section occupies 3 pages in my browser and is nearly a third of the body of the article! My full comments are at Talk:Derivative#This article has to do too much work, but I've moved the discussion to a new section rather than there so that my comments don't dominate the section. I think Rules for finding the derivative could be shortened and rewritten and a few standard derivatives could be given, with more prominant (main article?) links to Table of derivatives (which could probably be split into separate articles on rules and standard functions). There is also some reordering, restructuring and rewording that I would like to try. Geometry guy 14:08, 25 March 2007 (UTC)

I agree with shortening the notation section. We can keep the two most used notations, meaning Lagrange and Leibniz, shorten the Leibniz's notation section, and then cut out the entire notation section from this article and move it to a new article as Jitse suggests (and refer to this article at the top of the now shortened notation section). Oleg Alexandrov (talk) 14:43, 25 March 2007 (UTC)

And I would like a source for "Euler's" notation; I was taught that as Heaviside's. Septentrionalis PMAnderson 22:32, 25 March 2007 (UTC)

Progress reports

Okay, I have done most of the tedious work in moving out the notation section and sorting out related articles. I've moved the unsourced tag to the new article Notation for differentiation. I've done very little other rewriting here, but made a start. I think we should mention all the notations briefly. Geometry guy 00:28, 26 March 2007 (UTC)

I've now tidied up the articles on notation a bit, and shortened the section here, as suggested above. The Leibniz notation subsection is still quite long, but I think that is justified as it is probably the most common notation in elementary calculus (even if it somewhat flawed from a more advanced modern point of view). I am inclined to reorder so that it comes first. I have also made what I hope are the least controversial of my other proposed changes, which involve grouping the applications together. Geometry guy 20:26, 26 March 2007 (UTC)

(Reordering done.) After creating a new Differentiation rules article (a fine slogan!), I edited the material here and at Table of derivatives and fixed many links. Today I moved on to some more challenging edits ;-). I spent some time rereading the edit history and the talk history: my feeling is that this article has been a victim of its own history. At some key moments when a good editor (e.g. User:Dmharvey) could have refreshed the article, instead another editor made an incomprehensible mess which had to be reverted and cleaned up. So I decided to rewrite the first section completely, to eliminate the redundant information coming from the "differentiability" point of view and the "Newton's difference quotient" point of view. I tried not to lose any previous editors' insights in this edit, but my main hope is that this is now more accessible and structured for the novice reader. I also incorporated some material from the calculus article. Geometry guy 01:45, 29 March 2007 (UTC)

I think that the article is in a good shape now, thanks to the recent almost single handed efforts from one determined editor. In my opinion, the first section is very well written, and frankly, I am not convinced that the Calculus article is "non-technical introduction" to that. Also, the overall organization of the material seems good to me. I do have a couple of concerns, but they are related to the technovisual aspects.

• Is it possible to display the table of content in two-column format? Currently, it occupies almost an entire screen, and makes it hard to glimpse at the first section from the introduction.
• I know that this had been discussed before, but I will bring it up again: is there a better solution than posting the list of calculus topics on the right of the lead? Especially in this case, at least on my browser all the lead text appears very ragged as a result. It's not at all a trivial issue, because the blue bands in the topics panel distract too much from the lead.
• Perhaps, the two questions can be addressed together by moving the topics down a bit to make it abut the TOC instead of the lead. If there is anything that should go on the upper right, it'd be some graph like Figures 1-3: sufficiently small not to cause the current problems, pleasant to the eye to increase friendliness (does anyone seriously think putting a crib-sheet or exam-topics-list-style bar makes the article more attractive?), but not too catchy.
• Finally, another question to technosavvy editors: can someone, please, archive the beginning of this talk page, e.g. to around the GA award discussion? It's insanely long at the moment. Arcfrk 04:24, 29 March 2007 (UTC)

Thanks! Unfortunately, I'm not sure what to do about the technical aspects. I moved the calculus banner down a paragraph, which is not much better, but helps a bit. More radically, it could be moved to the "Computing the derivative" section - opinions? As for the table of contents, I don't know a fix. I am partly to blame for expanding the number of subsections, but I feel these add to the readability. Anyway, I reduced the number of subsections by one.

It would be good to have feedback from a (non/less)-mathematical editor about the changes. I believe I have made the material more accessible than it was before, while maintaining (perhaps even enhancing) an encyclopedic tone — but as both a mathematician and the contributing editor, it is hard for me to be objective!

I also expanded the history section, in line with the post-GA recommendations. Geometry guy 19:03, 29 March 2007 (UTC)

I have now completed the last main task to address the issues I raised on 23 February by adding a section which is an introductory survey on derivatives in higher dimensions. This material is, of course, intrinsically more advanced than the one dimensional derivative, but I have done my best to make it as accessible as possible by relating the various notions to the one dimensional derivative and to each other, and also by providing clear links to articles that elaborate these ideas. There is still plenty of scope for improvement, though, I am sure! As I mentioned before, as long as this article is the main article on derivatives, I think it is essential that it should address the central concept of multivariable and higher dimensional differentiation in as elementary way as possible.

I have also rewritten the "generalizations" section in the light of this addition. I would also like to tidy up the "applications" section (which is not entirely encyclopedic right now) and update the introduction. I will have a go at this shortly. In the long run, it would be nice to have an Applications of differentiation article. And of course, it is necessary to make sure this article is well-sourced... Geometry guy 16:56, 31 March 2007 (UTC)

Progress?

With a redraft of the introduction, the rewriting I wanted to do is done. The article is no shorter than before, but I believe it is richer in content and branches out to several improved supporting articles. My fondest wish is that those who liked the article before still do, and those who didn't, do now. Of course (WP-)life is not that simple, so long may the comments, discussions, criticisms and edits continue! I would really like to see this article reach A-class: it is such a fundamental topic in mathematics that it is a great pity that it isn't A-class yet. Geometry guy 18:38, 31 March 2007 (UTC)

Recent edits to intro and first section

141.211.120.199 (aka 141.211.63.35) has made some useful edits to the introduction and the first section. Some of them seem to me to be definite improvements, but I'm not completely convinced by all of them. I've only made fairly minor corrections to them so far. However, I do have a couple of requests for 141.211.120.199.

• Please do not use the word "copyedit" in the edit summary to describe an edit which adds to or changes the mathematical content.
• Please break your edits into stages so that others can see more easily what changes you have made using the diff's in the edit history.

This is just a matter of courtesy to other editors on wikipedia. Thanks! Geometry guy 13:01, 18 April 2007 (UTC)

Hmm. Most of the time, I'm not sure what else to say besides "copyedit". I'll try to be more specific, and I'll try to commit things in smaller chunks—but I should warn you that I'll be writing it the same way as I have, in big chunks, so it may happen that some of those individual edits will look strange in isolation. We'll see how it goes.

Also, I think I should give you an explanation, because you haven't seen me before. (Before this, I've mostly stuck to advanced articles like spectral sequence and sheaf (mathematics)—I'm rather proud of those two—and my only serious foray into more elementary topics, Riemann integral, didn't elicit any response.) First, I take the commandment be bold very seriously, and if I want to change something then I change it. If you don't like it, you should be bold and change it yourself. With an article as good as this one I'm changing mainly the exposition, not the content, and that's something that can be tinkered with endlessly. Second, I like my anonymity, and I want to remain anonymous even among Wikipedians. I'm well-aware that you can see my IP address and that an account will bring privileges I don't have. I'm still not getting one. (I seem to be an Exopedian.) I hope that clears things up a little. 141.211.62.20 15:14, 18 April 2007 (UTC)

Many thanks for the clarifications. I'll copy the second part of your message over to User Talk:Geometry guy and reply further there. I agree with you about being bold: if I don't like something about an article, I try to be bold and change it too. The only comment I am making is that if there are two or three different things I don't like about an article, I tend to change them one at a time. The more elementary and evolved an article is, the more likely I am to do this. I find that the easier I make it for other editors to see where I am going, the easier it is to improve the article, with less work for everyone (including me!). Anyway, this is just a suggestion, and you already do this to some extent anyway. I cannot and do not wish to change the way you work! Geometry guy 16:28, 18 April 2007 (UTC)

PS. (Over edit conflict with Salix alba.) I'll also move Salix Alba's reply to my talk page. Please revert if you have any objections to me doing this.

Question

Hi, isn't the line in the first picture function technically also a secant line; for it passes the peak and the trough simultaneously. Just wanted to clarify this VinceyB 19:28, 23 April 2007 (UTC)

Yes, it's also a secant line. This is a very common phenomenon to which the most important exception is quadratic polynomials. If you have a polynomial which is not a quadratic, there are usually lots of tangent lines which are also secant lines. If you consider intersections over the complex numbers, then every tangent line (to a non-quadratic polynomial) is a secant line. This is because of Bézout's theorem. 141.211.62.20 23:05, 25 April 2007 (UTC)

Oops, that's not true. For example, the tangent line at an inflection point of a cubic polynomial is not a secant line. (E.g., f(x) = x³ at x = 0.) However, it holds for all but finitely many points on the graph of the polynomial, the exceptions being the points where the tangent line meets the graph with multiplicity equal to the degree of the polynomial. That is, if you take the Taylor series of f(x) around a, and if that starts out f(a) + f'(a)(x - a) and has no more terms until the term of degree n, then the tangent line at a is not a secant line. 141.211.63.35 18:08, 27 April 2007 (UTC)

This article needs reorganization

I'm pretty happy with the text of the article at this point. We could all tinker with it ad infinitum, of course, but I think that, when looked at on the paragraph-by-paragraph level, the article is okay as it stands. (Though I've decided I don't like the way footnotes are used. I know I'm guilty of adding some. I repent, I won't do it anymore.)

What I'm not happy with is its organization. The article is scattered and only somewhat coherent. It starts off with the single variable case, then switches to notation, then switches back to the single variable case for computations. Then it generalizes to the higher variable case, switches to history and applications, then switches back to generalizations at the end. So while the article is find on the paragraph-by-paragraph level, I don't think it works on the section-by-section level, because the sections don't fit well together.

Here's a proposal. Tell me what you think:

• Break off most of the single-variable stuff into a new article, say derivative (single variable). This would absorb most of the first section. It could also absorb some, maybe even all, of the computation section.
• Change the focus of the explanations to the derivative as the best linear approximation. This is, of course, what the derivative "is", in the philosophical sense. There would be a section titled "The derivative as the best linear approximation", and its plan would look something like:
  • The derivative in single variable case. Leave out the stuff about the derivative as a difference quotient, but emphasize the derivative as the slope of the tangent line and as the best linear approximation. This has a geometric feel to it and can be easily explained through pictures (which, fortunately, we already have).
  • The derivative in the multivariable case. The "best linear approximation" stuff is built in to the definition of the total derivative, of course.
  • Other generalizations. The current generalizations section could be used as-is, though if we're going to focus on the linear approximation property it might want some touching-up.
• And everything else would go in other sections: Notation, History, Applications, Bibliography.

I think this would make a more consistent article than what we've got now. Thoughts? 141.211.120.130 23:15, 29 May 2007 (UTC)

Sounds good to me, but since it is a massive change I think that editors will want to see it first. Feel free to use my sand box to implement your idea User:Cronholm144/Derivative.Good luck!--Cronholm144 01:38, 30 May 2007 (UTC)

My how things have changed since February when this article only treated the one variable case! As can be seen from the comments I made then, I am personally a big fan of the "best linear approximation" point of view. However, as long as this is the lead article for Category:Differential calculus, I think it needs to be really accessible, and I am concerned that the proposed changes would reduce its accessibility. Also, while "best linear approximation" may be a good unifying theme for the category, overemphasising it in a given article at the expense of other important viewpoints (rates of change, difference quotients) may not be encyclopedic.

Apart from the location of the history and applications sections, the sections are currently in order of increasing depth, but I can see the case for a reordering such as

Definition, Computing, Higher dimensions, Generalizations, History, Notation, Applications

similar to that proposed by 141.211. An alternative (which is quite often recommended at WP) would be to lead with the history, although I am not a fan of this approach. It is also conventional to end with generalizations, but there is no reason to be bound by convention.

The thing I am most unsure about is moving the one variable stuff out. I'm not against the idea in principle, but it needs to be done for the right reasons. For instance, concepts such as higher derivatives, and rules of differentiation (especially the product rule) are fundamental to the idea of differentiation. If these are moved out so that the best linear approximation viewpoint flows together more seamlessly, then I think something may be lost.

Anyway, I'm not set against some restructuring, and Cronholm's suggestion to sandbox it first is a great idea. Then we can evaluate the pros and cons more carefully. Geometry guy 11:11, 30 May 2007 (UTC)

I will very strongly disagree with breaking off the single variable case. Let us keep in mind that Wikipedia articles should be accessible, and 95% of the people who will want to read the derivative article will want to see the single-variable case. Oleg Alexandrov (talk) 15:23, 30 May 2007 (UTC)

Rewrite available for feedback

OK, I think I've finished my complete rewrite. It's available at User:Cronholm144/Derivative. Many thanks to him for giving me a sandbox. This is such a big rewrite that I don't know if I could have done it without that. Here are some notes:

• I decided to write "pushforward" where a differential geometer might have said "differential", and I adopted the pushforward-like notation f_* rather than the differential-like notation df. I did this because df looks like the differential form df and like something coming from the Leibnitz notation. The pushforward isn't the same thing, so I thought that using df would be confusing.
• In light of an article referenced on the Calculus talk page, (Conceptual Knowledge in Introductory Calculus, Paul White; Michael Mitchelmore Journal for Research in Mathematics Education, Vol. 27, No. 1. (Jan., 1996), pp. 79-95 [1]) I decided to avoid using variables for constants. Saying "Fix a constant a" might make us happy, but apparently it does not make students happy. (Strangely enough, I remember the moment when I realized that some variables were fixed and some were variable. It must have made quite an impression on me.) Consequently you see actual numbers everywhere. This can be changed if we agree it's not a good idea.
• The proposed article Derivative (single-variable) would contain most of the material that was deleted: More detail on the difference quotient, a worked example, and comments about continuity. There's additional stuff which can go there; for instance, apparently there are some interesting restrictions on what functions can possibly be derivatives of some other function.
• Finally, as with any draft, there are probably solecisms lurking. There are places where the formatting can be improved. Please excuse my errors.

Moving forward, there are four paths I can see from here:

1. Adopt the rewrite mostly as it is.
2. Adopt parts of the rewrite, adapting them to fit the current article.
3. Significantly modify the rewrite, but keep the same general structure and outlook.
4. Throw out the rewrite.

As you might expect, I'm partial to the first option, but this is a team effort. Opinions? 141.211.120.86 00:24, 10 June 2007 (UTC)

Wow, nice work. I don't know if the pushforward stuff belongs in the main article though, although I agree that from a mathematical point of view, it's much better to think of the derivative as a linear map. By the way, the differential is the same as the pushforward in the sense that if f : M → N is a mapping of manifolds, then f_* = df : TM → TN is a linear map of vector bundles covering f. When N = R, you recover the usual differential form df you're thinking of. As for what to do with your rewrite, I'm not sure. May I make another possible suggestion:

• Incorporate the new changes with one or both of the existing articles linearization or total derivative (the relevant redirect from differential (calculus)).

To me this seems like a more reasonable solution than adopting a comparatively advanced viewpoint in the derivative article. Thoughts from other editors would also be appreciated. Silly rabbit 08:54, 11 June 2007 (UTC)

Thank you. It's great to hear that at least one person thinks the rewrite is nice!

You're right, pushforwards and differential forms are the same. (I usually think of differential forms as sections of T*M.) But the Leibnitz notation is a subtle thing, and using both df and dy/dx suggests that you can divide differential forms; I don't know how to explain that that's no good in elementary terms. You may have thought more about that than I have, and if you can do it, it might make a good addition to this article or to some other article. (On the other hand, the distinction between a differential form and a derivative is important, and if we consistently use df then that sets the reader up to integrate differential forms. Maybe that outweighs the possible confusion? The notation would get prettier: df_{(5, -2)} versus f_*(5, -2).)

I don't think the viewpoint is as advanced as it appears. I don't wait until the total derivative to talk about linear transformations, so it does look a little more advanced. But most of the rewritten material doesn't really use linear transformations or vector spaces, the big exception being "The single-variable derivative as a linear transformation", and even the parts that do use mostly one-dimensional vector spaces. The main difference from the present article is the emphasis on the derivative as the best linear approximation, which, I think, comes closer to capturing the soul of the derivative than the present article's technique. That was what I really wanted to capture with this rewrite, but I found that I couldn't avoid talking about linear transformations without doing the material a serious injustice; if we left linear transformations out, how could we sleep at night? 141.211.120.108 15:54, 11 June 2007 (UTC)

If there is a way to introduce the linear approximation (or linearization) approach to differentiation here in an unobtrusive way, then I think the article would benefit. The main advantage is that this gives a uniform way to think about derivatives as geometrical objects. The disadvantage is that, in an article intended for the masses such as this one, organizing the article around this notion is going to lose most of the prospective readers. (The phrase pearls before swine comes to mind.) The existing articles on linearization and total differentiation are in a sufficiently poor state that I would suggest incorporating some of this material there. The linear-approximation approach to the derivative could then be summarized here (see WP:Summary style), with a {{main|Linearization}} or somesuch link template. Silly rabbit 21:35, 12 June 2007 (UTC) PS: I note the deafening silence of other editors. Does anyone else have any thoughts?

I think it is good content but agree with SR that its probably not appropriate full scale in this article. Maybe there is a new article where it could go, modern approach to calculus or some such title.

One concern is that it is really discussing tangent spaces without explicitly mentioning them, so in some respects its not technical enough.

I don't know if this is a sill question but how does del fit in this framework? --Salix alba (talk) 22:18, 12 June 2007 (UTC)

Re: User:Silly rabbit. Introducing the linear approximation approach was exactly my intent. User:Geometry guy said above that he is a "big fan" of that viewpoint. User:Dmharvey also seemed to be in favor of this approach (see Talk:Derivative/archive1). You say you think the article would benefit. And, as I said above, I think that "best linear approximation" is what the derivative really is—hence my desire to make the article reflect that.

"Slope of the tangent line" seems to go over better in class than "instantaneous rate of change". One semester I had a class really get hooked on the tangent line approach; it was hard to get them to compute algebraically, because they liked drawing tangent lines so much! Rather than being "pearls before swine" I think that our readers will appreciate the fine pearl of linear approximation.

Re: User:Salix alba. I really do intend this as an introduction, and not as a modern approach—that would also be good, but it's a different article. I agree that it would be nice if our readers understood tangent spaces, but because we're working on open subsets of Rⁿ we can, and should, avoid them. Almost all of the proposed article doesn't require that the reader know anything about vectors, and the parts that do only require a very modest knowledge. We could even segment off "The single-variable derivative as a linear transformation" into a subsubsubsection which began with a warning that this part, and the other mentions of linear transformations, is necessary only for understanding the total derivative.

Regarding del, I believe that del is another way of writing the exterior derivative. This means that it is a connection. The one-variable analog is to take the differential form df associated with f; since df = (∂f/∂x)dx this is another way of approaching the derivative. However I should defer to Geometry guy, who is actually a differential geometer and understands these things better. 141.211.120.77 23:30, 12 June 2007 (UTC)

Apologies for the silence. I wanted to step back a bit and let others comment, because the current structure has a lot to do with my handiwork, and so was concerned to be too close. However, the discussion has died down, so it is time to add my view.

I think 141.211 has produced a lot of good material here, but I am not sure what should be done with it, and I stand by some of the concerns I raised (about this being the main article for Category:Differential calculus and about the need for encyclopedic content) before the draft was written.

First, though, some replies to the above.

• del in the sense of nabla is indeed a standard notation for a connection on a vector bundle. This is different from the exterior derivative although the exterior derivative on functions is a connection on a trivial bundle. I guess nabla is more notable to most readers as a major tool in vector calculus to express div, grad, curl and all that. Its use for the gradient is of course closely related to the exterior derivative on functions, except that the gradient of a function is viewed as a vector field (using e.g. a Riemannian metric) whereas the exterior derivative is a 1-form.
• del in the sense of ∂ is rarely used for the exterior derivative, although it is sometimes used for the "complex linear part" of the derivative, as a counter-part to the del-bar or Cauchy-Riemann operator (which is the complex antilinear part). It is of course, also used for boundary operators in topology and homological algebra.
• In one variable you can take ratios of differential forms because the vector space of differential forms at each point is one dimensional. This is consistent with Leibniz notation, although explaining this to a general reader is, admittedly, somewhat challenging!

Second, a couple of minor things. I agree with comments above that using f_∗ for the derivative as a linear map is not a good idea at this level. I'm not a fan of the overarrow notation for tangent vectors (and bold font is used in a couple of placed). I also don't like all the explicit numbers: even if it is good pedagogy, I find it unencyclopedic (and a bit cluttered).

Which brings me to the general issues I mentioned before. It is interesting to note that the context for the aforementioned remarks of Dmharvey very much concerns the challenge of making encyclopedic material approachable. It is like teaching with your hands tied behind your back (no handwaving!). It is a real pain because, for instance, standard good teaching practice such as "Don't introduce a second point of view until students have mastered the first" often conflicts with encyclopedic principles.

The problem for me is that Derivative is (still) the main article in Category:Differential calculus, and so it has to be a scholarly reference piece as well as being accessible. 141.211 done a fantastic work to explain the derivative as a linear map in an elementary way, and there could be a great article on Introduction to differentiation as linear approximation (or something like that) in the making here, but at the moment it seems difficult to incorporate this into Derivative while maintaining a good balance of content, encyclopedic rather that textbook style, accessibility, NOR, NPOV and all that jazz.

I am a big fan of the linear approximation point of view, but this appears in three ways initially: first, there is the tangent line, then there is the derivative as the slope of the tangent line (which is subtly different, because it does not locate the line), and finally there is the derivative as a 1x1 matrix or linear map. I agree that the tangent line and linear approximation is a great way to introduce and explain the derivative, but would question whether one can throw all three of these variations at the reader so quickly, although 141.211 has given it a pretty good shot!

I suspect the issue again is partly that Derivative still has to do too much work. When I did some rewriting a while back, I was surprised, having read the talk archive, that the rather sweeping changes I made were virtually uncontested. It may have partly been because I developed a couple of subarticles first, so that when I made changes here they were clearly seen as steps in the right direction. I'm not sure what the right direction is at the moment.

This might become clearer if some related articles such as Linearization, Derivative (generalizations) and Total derivative are improved. Maybe it is time to attempt an overview at Differential calculus (which currently redirects here). Maybe we should try to write Derivative (single variable): some of 141.211s material would certainly be useful for that! I favour implementing the idea of linear approximation as a unifying idea for the entire category, before deciding how best to add it to the main article.

So my view is that it would be good incorporate some of 141.211s ideas here, but not all of it, at least not yet. Some of it may also be useful for related articles, and I would favour revisiting the issue at a later date. However, this is not www.geometryguyknowsbest.com, so I leave these rather long thoughts for other editors to consider. Geometry guy 16:59, 23 June 2007 (UTC)

Let me try to summarize everyone's remarks before I go on. If I misinterpret or err, please correct me:

• The article Derivative is currently both an introduction to both differential calculus and all of its trappings (optimization, differential equations, etc.) as well as an introduction to the derivative operator.
• The article I produced is heavily weighted towards the derivative as an operator and does not provide a balanced introduction to differential calculus.
• The article I produced is too advanced to serve as the main article for Category:Differential calculus.
• Related articles such as Linearization, Total derivative, and Derivative (generalizations) need to be developed.

Assuming that's a correct summary, I think I know what needs to be done.

• Develop some of the articles in Category:Differential calculus.
• Write Differential calculus, a new main article for Category:Differential calculus which explains the derivative in vague terms and puts a lot of emphasis on its applications.
• Remove from Derivative anything not immediately relevant to the derivative operator.

My vision is that Differential calculus will contain things like history and applications (to differential equations, physics, optimization, etc.), while Derivative will contain things like the definition, rules for computation, and so on. The really lacking thing, and the whole reason why Derivative has been doing too much work, is the absence of Differential calculus. This should include at least:

• A non-technical overview of the derivative.
• History of differential calculus.
• Applications:
  • Physics
  • Chemistry (e.g., reaction rates)
  • Optimization problems
    • Calculus of variations
    • Operations research
    • Game theory
    • Pattern matching (e.g., minimizing energy functionals in computer vision)
  • Related rate problems
  • Taylor series
  • ODEs and PDEs
  • Implicit function theorem

I'm sure there are lots of other applications out there—I only spent a few minutes looking around to get all of that. Assuming that we had an article like that, it should be easy to adapt either the present derivative article or my draft to be the new derivative article—cut the history and applications and write a new lead. (Either one would be better if they meshed with expanded articles on linearization and so on, but they're both good first approximations. (Linearizations! :-) )) What would everyone think of that? 141.211.120.87 17:44, 23 July 2007 (UTC)

It has become very quiet around here! The vision you suggest sounds very good to me. The only thing I would add is that Differential calculus should contain the overarching idea, that differentiation is about linear approximation, explained in an elementary way. Geometry guy 09:51, 27 July 2007 (UTC)

Okay, I've made a first start. It's rough and incomplete, but it's there. 141.211.62.20 03:28, 28 July 2007 (UTC)

Picture

I made a pic illustrating a function and its derivative (I was thinking of x^2/3 when I drew it) anyway I noticed that Kingbee had a problem with the current pic so here is. BTW things sure have gotten quiet around here since this article made GA...--Cronholm¹⁴⁴ 15:03, 25 June 2007 (UTC)

I think the problem kingbee had was with the caption, which gave the misleading impression that continuity implied differentiability. (In fairness to its author, it did not say this explicitly, but rather left out an important qualification.) I have added an image of the absolute value function to the Continuity and differentiability section, since the text deals with the absolute value as well. Silly rabbit 15:53, 25 June 2007 (UTC)

Thanks Silly rabbit! ... I feel strange typing that... Oh well that was a good addition, it is important to "differentiate" between necessary and sufficient conditions...I obviously need to go to bed if I am make puns that bad...and if I am talking to myself...sigh...ending post. --Cronholm¹⁴⁴ 16:02, 25 June 2007 (UTC)

GA on hold

This article has been reviewed as part of Wikipedia:WikiProject Good articles/Project quality task force. In reviewing the article against the Good article criteria, I have found there are some issues that need to be addressed, particularly references. Many equations are not referenced, and some of the simple equations should be relatively easy to find (in a book or on website). I am giving seven days for improvements to be made. If issues are addressed, the article will remain listed as a Good article. Otherwise, it will be delisted. If improved after it has been delisted, it may be nominated at WP:GAC. Feel free to drop a message on my talk page if you have any questions. (oldid reference #:153229464) OhanaUnited^{Talk page} 02:40, 24 August 2007 (UTC)

The article is now on GA/R. OhanaUnited^{Talk page} 21:24, 8 September 2007 (UTC)

I understand your concerns about referencing, however in this article there is a reason why the formula are not referenced, they are all very common appearing in every text book. More to the point the references given at the end are more than adequate for the reader to verify the formula. The reader is better served by providing a a range of references. I can't see how it wily help to add lots of links to the same book. --Salix alba (talk) 22:40, 8 September 2007 (UTC)

I concur. Inline references for every mathematical statement in every mathematical article on Wikipedia would be excessive and unnecessary (and would not reflect the style of works in the field). Points that can be verified in standard texts need only be referenced by a reference at the end. Inline references would be suited for claims of particular notability or authorship, or that are non-standard in some way. --Cheeser1 12:52, 14 September 2007 (UTC)

Paring external links

There are currently 4 derivative calculators in the external links. I think it would be best if we could just pick one or two and remove the others. Any suggestions as to which to pick? —Cronholm¹⁴⁴ 08:01, 14 September 2007 (UTC)

Well, I figured a test was in order. I input the function e^(2x+sin(x)) into each and got:

e^(2x+sin(x))* (((2x+sin(x))*0)/e+ (cos(x)*1)*log(e)) from Solve My Math

f'x = (2+cos(x))*e^(2*x+sin(x)) from Online Derivatives Calculator

a correct graphical representation like this* from WIMS Function Calculator

Syntax Error from ADIFF (I honestly can't get it to work - the syntax here is notably over-complicated)

In light of this, I'd say the Online Derivatives Calculator and the WIMS Function Calculator should stay. I've gone ahead and removed the other two to reflect this. *Warning - unreliable server (sorry!). --Cheeser1 12:50, 14 September 2007 (UTC)

Someone put the ADIFF calculator link back in, so I removed it again. It is very clunky and inelegant, and we already have other derivative calculators on the page that are nicer. By default ADIFF doesn't know what to do with sin x, for instance. As a result, this doesn't seem useful for the general reader (see the guidelines for external links at WP:EL). Doctormatt 17:08, 3 October 2007 (UTC)

Adiff is a general purpose derivative solver. People have been using it. To enable Trig functions, it is only necessary to refer to Example 1 on the page. I provide a bulletin board, and a help page for people to ask for further assistance ufnoise. —Preceding signed but undated comment was added at 20:07, 3 October 2007 (UTC)

I've also run the provided example in this forum, and I get the correct answers inputing the following.

declare(cos(x))

define(sin(x),cos(x))

define(cos(x),-sin(x))

diff(exp(2*x+sin(x)),x) —Preceding unsigned comment added by Ufnoise (talk • contribs) 20:12, 3 October 2007 (UTC)