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Simple English?

The language in this article still baffles me. In all honesty, calculus in general baffles me! Still it would be nice if someone could dumb it down enough to write up a Simple English Wikipedia version of the article - (Simple:Derivative) .. if that's even possible. -- œ 03:22, 4 April 2010 (UTC)

    It is an easy mistake to think that Simple is intended for users are not prepared with the underlying knowledge needed for understanding a particular article in this WP. Rather, its purpose is to serve non-native speakers of English whose English vocabulary is small, while presuming the same level of underlying knowledge appropriate to the corresponding topic in any of the other languages of WP.
    The article you want would be on this WP, with a title like, perhaps, Minimal concepts for understanding differential calculus. I'm not sure we have any distinct articles with this role, and i'm pretty sure there is no template along the lines of {{prerequisite}}. We may need a WikiProject intellectual accessibility (with sub-projects that each have a respective major subject area as another parent) to review the lead sections of the more technical articles woth the goal that as large as feasible a fraction of users will recognize which links they need to follow to get up to speed for the article itself.
    OTOH (and more likely why i haven't seen those in the last 7 years -- rather than bcz of their existing under other names), the ability to tackle a given article depends importantly on at least two factors about the reader: cognitive style, and background knowledge (which probably has, as its primary determinants, the content and level of formal education or intensive self-directed study, and experience from work and "hobbies"). Its not unreasonable to argue that an encyclopedia is not a textbook, and can't reasonably hope to serve the needs of, say, someone who wants to understand differentiation but hasn't previously learned advanced algebra (if i correctly recall the title of what i once took). I recall being convinced (during the course that i mean) that i had absorbed the concept of function (mathematics), but going around for an extended period w/o being able to grasp why it was worthwhile to single out that concept for a formal definition; that memory leaves me suspecting that
  1. anyone who first encounters the concept of "function" in WP will need a textbook -- if not an instructor -- in addition to WP, to understand any plausible derivative encyclopedia article, and
  2. anyone who first encounters "derivative" here will need a textbook or instructor to understand any plausible partial derivative encyclopedia article.
(Someone remarked to Charlie Rose the other night that the only institutions that have survived the last 500 years unchanged in their essential nature are universities. So the inherent structure of knowledge, rather than the tendency of privilege to be used to preserve privilege, is probably the explanation for the academic system -- whatever media theorists may speculate.)
--Jerzyt 20:02, 4 August 2010 (UTC)
Thanks for that.. I actually did check out the Wikibooks links to the various textbooks, but they weren't much help either.. But I understand what you're saying.. maybe if I actually bothered to pay attention to high school math lol! -- œ 15:19, 5 October 2010 (UTC)
The purpose of this article isn't to teach calculus, but to describe what it. This description may involve complex vocabulary and mathematics; it's not meant to pander to beginners JDiala (talk) 19:18, 15 December 2013 (UTC)


Can someone explain what the hell is going on with the Kim Hyun Bin nonsense? Only dead fish go with the flow. 17:52, 16 May 2010 (UTC)

Actually, you know what, can someone do something about whatever has been doing? Someone with rollback? Only dead fish go with the flow. 18:18, 16 May 2010 (UTC)

Picture of function and derivative

It would be nice I think for the picture showing the tangent at the begining to show the graph of the derivative function as well.

An idea for linking the two might be to show a triangle base length one on the x axis with the same direction hypotenuse as the tangent line the top point would move along the graph of the derivative. Dmcq (talk) 12:07, 18 June 2010 (UTC)

derivative in physics

If x is time t, y is speed v and if c is acceleration a, then line y=c*x=v. So x=t, y=v, c=a. And y'=(c*x)'=c*1=c. So speed v derivative v'=a. And v=t*a, v'=a. Or more precise can be written v'(t)=(t*a)'=a. Or . Seems that no more else derivative in physics is used related with acceleration, speed, time or distance.

HOW CAN I KNOW CALCULUS? —Preceding unsigned comment added by (talk) 12:54, 19 October 2010 (UTC)

If I interpret you correctly, you seem to be saying that the page on derivatives should point out the connection to physics, namely derivative=speed and second derivative=acceleration. This is a very good point. The relation to physics should certainly be discussed, as it is of great benefit to all, and particularly to beginners. Tkuvho (talk) 13:12, 19 October 2010 (UTC)
A more general applications section may be more useful. It could include, for example, the reaction rate and rate equation of chemical kinetics, in addition to the the example listed above. Mindmatrix 15:58, 19 October 2010 (UTC)
Personally I think this is a major oversight indeed, but perhaps this is treated at a different page? Should we bring this up at WPM? Tkuvho (talk) 16:08, 19 October 2010 (UTC)
Sure. Perhaps a more generic discussion about the inclusion of applications in articles about various mathematics concepts is warranted. The project may have already had such a discussion many years ago (perhaps 2005?), but I don't recall with certainty. One of the probelms with including applications is the eventual laundry list of every application being added, overwhelming the article. One possibility is to create a List of applications of derivatives et al. Mindmatrix 17:14, 19 October 2010 (UTC)
Bah - I completely forgot about Differential calculus#Applications of derivatives. Sigh. Mindmatrix 17:16, 19 October 2010 (UTC)

Wrong Derivative in animated Gif

The derivative that appears in this picture is wrong. It should be:

But I don't know how to modify this animated gif! —Preceding unsigned comment added by (talk) 12:06, 14 December 2010 (UTC)

You are incorrect. --Izno (talk) 14:35, 14 December 2010 (UTC)

Section "Partial derivatives": incorrect formula?

The section "Partial derivatives" has as the second-last formula


This seems to me to be incorrect. Looking at the notation in the equation preceding it in the section, I think it should rather be


Any comments? I will leave this edit to those more familiar with the process, unless asked to do it. Quondum (talk) 09:16, 27 December 2010 (UTC)

I've fixed it. Thanks! Ozob (talk) 11:55, 27 December 2010 (UTC)

Symmetric derivatives

An anon has recently been trying to add the symmetric derivative here as an alternate definition of the derivative. For everyone's benefit, I would like to prove that the symmetric derivative is not equivalent to the usual one. I'll give three counterexamples.

First: Let f(0) = 1 and f(x) = 0 for x not equal to zero. That is, f is zero everywhere except at 0, where it jumps up to the value one. This is not differentiable at x = 0 because it's not continuous. But it does have a symmetric derivative:

This is because when we take the limit, the value of f when h is zero is irrelevant. Because of this, the symmetric derivative never notices the discontinuity at x = 0.

Second: Let f(x) = |x| be the absolute value function. f is continuous at zero, unlike before, but it is not differentiable there because its one-sided derivatives are not equal:

Its symmetric derivative isn't equal to either of these:

That is, the symmetric derivative does not notice that the one-sided derivatives disagree. Whenever both one-sided derivatives exist, then the symmetric derivative is their average.

Third: Let f(0) = 0 and f(x) = xsin(1/x) for x not zero. f is continuous at zero because |f(x)| is bounded by |x|. Neither one-sided derivative of f exists at zero:

which does not exist; similarly for the other side. But f does have a symmetric derivative at 0:

So the symmetric derivative does not notice that the one-sided derivatives do not exist.

I hope this clears up any confusion there might be. Ozob (talk) 01:46, 4 April 2011 (UTC)

I have just a minor issue with your math, and it is the third line:
How is abs(h) equal to -h?... --Izno (talk) 15:29, 10 May 2011 (UTC)
In that limit, h is approaching zero from below. That means that h is negative, so |h| = −h. Ozob (talk) 01:21, 11 May 2011 (UTC)

a useful formula

Jackzhp (talk) 14:47, 10 May 2011 (UTC)

I agree that this is a useful formula. It appears (with different notation) at Differentiation under the integral sign. Do you think there is a good place to mention it in this article? Ozob (talk) 01:24, 11 May 2011 (UTC)

Petition to reorganize a sentence in the first paragraph.

The sentence in question: "Conversely, the integral of the object's velocity over time is how much the object's position changes from the time when the integral begins to the time when the integral ends." is abrupt and has no direct connection to derivatives.

Suggestions received happily. —Preceding unsigned comment added by (talk) 20:00, 19 May 2011 (UTC)

I've never liked that sentence, and for the same reason (that it has no direct connection). I think we ought to delete it. Ozob (talk) 12:20, 20 May 2011 (UTC)

Nonstandard Calculus

Should we also introduce the nonstandard approach here? Dalcde (talk) 09:14, 1 August 2011 (UTC)

I'd prefer not to discuss it in detail. The definition by difference quotients is, nowadays, the standard one, and giving a full exposition of other approaches would give those approaches undue weight. But right now the article mentions other approaches only in a single footnote, and that might not be enough. I think it's worth having a sentence or two on other ways of defining the derivative. Ozob (talk) 11:22, 1 August 2011 (UTC)

Apostol's use of "in the direction" vs "directional derivative"

I don't have a copy of Apostol's book, but based on controversies I've seen on the web, can you clarify whether he actually uses the term "directional derivative" when defining a derivative that is taken with respect to a general vector v? Or does he use the terminology of a "derivative in the direction v"? There are some sources (for example the Wolfram site) that reserve the term "directional derivative" to refer to the case where v is a unit vector. Does Apostol's definition contradict this use of the term "directional derivative" or does he avoid using the term "directional derivative"?

Tashiro (talk) 15:03, 5 September 2011 (UTC)

section needed for the formal definition of a derivative

The information here looks terribly disorganised and could do with tidying up. I was thinking of a section that can give a formal definition using modern maths. The rest of the article is ok, it just seems to be more suited for high-school children. — Preceding unsigned comment added by Jmc2000 (talkcontribs) 01:34, 5 August 2012 (UTC)

Formal Definition unclear

Currently, text is "The derivative of y with respect to x at a is, geometrically, the slope of the tangent line to the graph of f at a..." This presents 'graph of f' as a curve, which exists at point a. Continuing "...The slope of the tangent line is very close to the slope of the line through (a, f(a))..." a is still a point, and f(a) is the same as 'graph of f" at a. While you can write (a, f(a)) like coordinates, it only means f(a) around a. Yet now the slope is different that 'graph of f' at a (though close.) Continuing "...and a nearby point on the graph..." A nearby point on the graph? That treats (a, f(a)) as coordinates. Con't "... for example (a + h, f(a + h))." Sure enough, f(a+h) is a coordinate.

The problem is that when the writer says 'slope at a,' a is an x,y point. When he writes (a, f(a)), it first continues to refer to a as an x,y point, but later a seems to become only a point on the x axis. How else to interpret '(a + h, f(a + h))'?

BrianMC — Preceding unsigned comment added by (talk) 23:04, 29 January 2014 (UTC)

I agree that the article was not completely specific about the tangent line. Is the current revision better? Ozob (talk) 02:45, 30 January 2014 (UTC)

Merge Higher derivative here

A recently forked article higher derivative has sprung up. I don't feel that there is enough new content to warrant a separate article, nor do I feel that there will ever be enough content to warrant such an article apart from what can be covered here and in third derivative, fourth derivative, etc. I thus believe that the new article should be redirected back to the section Derivative#Higher derivatives. Opinions? Sławomir Biały (talk) 22:49, 4 October 2012 (UTC)

Higher total derivatives section ending

The section about higher total derivatives ends with a somewhat vague paragraph saying that higher differentials are not linear transformations and references the page on jets. However, it the second example on that page assumes knowledge of the concept of higher-order differential, which is not explained there, as far as I can tell, on that page. I think the generalization that is intended in both cases is the Fréchet derivative (in the special case Rn → Rm), but I'm not sure as the Fréchet derivative is a linear transformation (from the n-fold product space of the domain, which is how it appears to be used in the jet article). I think it should be made clearer just what that last paragraph refers to, because it doesn't seem like it's referring to jets or any other concept I can find. Cyrapas (talk) 20:33, 14 December 2012 (UTC)

Longest Derivative Solution By Sandeep D. Sontakkey

Hell, There is new developing ideas in solving the Problems in Derivative............................ thank u..............

See u soon............ — Preceding unsigned comment added by (talk) 14:01, 2 January 2013 (UTC)

Continuity: differenciation & integration, step function graph.

The function does have a derivative at the marked point, you need to change the axis of observation to obtain that derivative. Pertinent, that a straight line results in a function delta(x)/delta(y), where delta(y)=0, which is indeterminant solely due the form of observation. The other aspect, where delta(x)=0 (or a constant), is known to lead to a result of zero, delta(x=0)/delta(y). Mathematically, NEITHER division nor multiplication by zero are allowable operations, both leading to one to many, or many to one solutions, which are solely solvable using gaussian functions. That particular function, is defacto continuous, by rotating the axis of observation slightly so that there are no 0/y nor x/0 aspects.

Leads to an interesting observation. Is a constant continuous, being it so that a constant has a derivative=0? The answer is, no, a constant is NOT continuous, it is bound on both sides by a sudden jump (a constant is a dirac delta), and is therefore not differenciable by definition. Because a constant is definitely differenciable (differencial of zero), all such sudden jumps are differenciable, and the platform itself, on the other side of that junction, collapses onto the axis. What does that imply? It implies that you are required to resolve the instance of discontinuity using a double axis, which is similar to using two graphs, one superimposed onto the other, whose resultant is that graph (math isn´t physics).

Someone needs to change the incongruities, errors, in relation to that function. — Preceding unsigned comment added by (talk) 12:08, 26 February 2013 (UTC)

This statement both has nothing to do with the article and is wrong. — Arthur Rubin (talk) 20:59, 26 February 2013 (UTC)

Proposed addition to section on higher derivatives

I would like to tentatively propose an addition to this article in the section on higher derivatives. My reluctance to simply edit the article is due in part to a conflict of interest: I am the author of the publication from which the proposed addition comes. Also, I am not a mathematician by training, so I seek feedback from professional mathematicians to assure that this addition is sufficiently significant and appropriate for an encyclopedia entry.

The addition I propose is a generalized equation for the computation of the nth derivative. It is the expression for f(n)(x) on page 387 of this reference. Subsequent to publishing, I learned that the difference functions, DIF(n,j), which are the subject of this article, are actually based on the Thue-Morse sequence. So the notation in the equation would have to be altered to conform to the accepted notation for the Thue-Morse sequence.

An argument for including this equation in the article on derivatives is that this compact general equation might prove useful. An argument against including it might be that this equation is not included in textbooks or in widespread use. A response to that argument might be that this equation may be undiscovered because it is not published in a math journal and omits reference to the Thue-Morse sequence, so Wikipedia would perform a service by identifying it. A response to that argument might be that Wikipedia is not the the proper forum to bring an unknown equation to the attention of mathematicians (then what is?), or that it is flawed (please state how), or that it is not significant (please state why). I would be grateful for any responses.

Robert Richman — Preceding unsigned comment added by Rmrichman (talkcontribs) 19:39, 27 February 2013 (UTC)

Suitable for Wikibooks, perhaps. If not published, it's not really appropriate here, per WP:OR which shows why the first two arguments against are appropriate. And I'd use the central formulation, instead; rather than
But, even so, the even more compact
could be used. Potentially, the first equation could be written:
but I don't really see the benefit. — Arthur Rubin (talk) 20:42, 27 February 2013 (UTC)

Problem for Q being the empty set

Let f: {a} \to R for some a \in R. f is then a real-valued function from a subset of the reals to the real numbers, in compliance with the definition in the section. According to the Weierstrass definition of continuity (as well as the sequential limit definition), f is also a continuous function.

Then Q is the empty set, Q: \emptyset \to R because for h \neq 0 f(a+h) doesn't exist and for h=0 there's division by zero. Any choice for Q(0) will make Q continuous at 0, by the same definitions as before.

So the question is, if Q: {0} \to R, does limit_{h \to 0} Q exist? Not according to the definitions of limit of a function on wikipedia (nor in my Analysis textbook), since 0 isn't an accumulation point. But suppose we were to say that the limit of Q at 0 equals Q(0). This means that f'(a) = Q(0), so not only does the derivative of a "one-point function" exist, but its value is the entire set of real numbers. That can't be right either. — Preceding unsigned comment added by (talk) 11:01, 15 August 2013 (UTC)

I thought that the derivative of a function on X at a was only defined if a is a limit point of X. Does that solve the problem, or are we having trouble with the definition of "limit". — Arthur Rubin (talk) 20:30, 15 August 2013 (UTC)
I should have referred to the last few edits on this article's page, which was why i added this section on the talk page here. As it stands now we have an inconsistency between the articles on continuity, limit of a function, and derivative (this one). Let f be as above, then let
P = "f is continuous at a"
Q = "the limit x->a of f exists and equals f(a)"
The article on continuity states P=true. The article on limit of a function states Q=false. This article on derivative states P <=> Q.
I had edited this article to remove the P <=> Q statement, but another user disagreed, so instead of back-and-forth editing i made this section on the talk page. I think the P <=> Q statement is incorrect, continuity is a weaker condition than the existence of the limit (it doesn't have the 0 < |x-a| condition, which makes all the difference here), it's the implication P <= Q rather than an equivalence. (talk) 23:15, 15 August 2013 (UTC)
OK, I think I see the problem, in terms of a not being a limit point of the domain of f. In this case:
  1. If f is defined at a, then f is continuous at a.
  2. exists and is equal to all numbers.
  3. If f is defined at a, then the derivative of f at a exists and is equal to all numbers.
There's something wrong, here. I would say that none of those are defined, but we need to adjust the definition of "limit" (and the corresponding definition of "derivative") and of "continuous". — Arthur Rubin (talk) 01:31, 16 August 2013 (UTC)
(I'm the same user as above, i've created an account in the meantime) I checked my Analysis textbook on this (it's a Dutch textbook "Analyse 1" by prof. Johan Quaegebeur at KU Leuven). It defines continuity in terms of the Weierstrass definition (noting that the sequential limit definition could also be used), so f is continuous at a, consistent with the wikipedia article on continuity. It defines the limit of a function in the same terms as the topological definition on wikipedia's page on limit of a function (ie requiring a to be a limit point), so does not exist, which is also consistent with wikipedia's article. It has a theorem stating the equivalence of those two only for limit points, which is inconsistent with the statement on this page on derivative.
I think the culprit here is the unqualified use of the equivalence of continuity and existence of the limit, whereas this equivalence is only correct for limit points. I believe my original edit resolves the problem, at least in accordance with my textbook, but this may be one of those borderline cases due to different authors using slightly different definitions. When viewed according to the definitions used in my textbook the article on "continuity" is strictly speaking correct, since it only uses "if" and never "if and only if" in its definition in terms of a limit, but it is very misleading due to the placement in the article that one could interpret it as "if and only if".
My proposal would be to remove the statement in this article using the equivalence of continuity and existence of the limit as per my original edit, and change the article on continuity to use the Weierstrass definition as main definition and delegate the "limit of a function" definition to a theorem applicable to limit points.B01010100 (talk) 03:02, 16 August 2013 (UTC)
The page does not say that Q is meant to be defined on an arbitrary subset of the reals. The only time the word "subset" appears is much later, in the section on Jacobians, and there the article says "open subset". I think it would be perfectly acceptable for the article to assume that the function is always defined in a open neighborhood of the point in which we're differentiating. That would avoid this problem, and it's the intersection of the definitions available in the literature. Ozob (talk) 03:23, 16 August 2013 (UTC)
The section starts with "Let f be a real-valued function." So maybe if we change this to "Let f be a real-valued function defined on an open neighbourhood of a.", that would solve the problem?
I do still think we should change either the continuity article to point out that it's "if" and not "if and only if", or change the limit of a function article to remove the condition that it must be a limit point. Either on its own is used in the literature, but together they are inconsistent - it's picking one definition to use in one article and another to use in another, even though both definitions may be used they can't be used together if one is to be consistent.B01010100 (talk) 03:41, 16 August 2013 (UTC)
I changed the article; while I thought the context was clear, I guess it does no harm to be explicit.
I don't have a strong opinion on what to do about the limit of a function and continuity articles. Since there are several alternatives available in the literature, it's probably best to mention them all (with citations) and point out their relative merits. The fact that the literature is inconsistent makes it very hard for us to be consistent – as an encyclopedia, not a primary or secondary source, we're really not supposed to be picking and choosing definitions, we're just supposed to be reporting on what definitions other people use. Ozob (talk) 15:24, 17 August 2013 (UTC)
The literature as a whole is inconsistent, but each textbook on its own isn't (at least any well-written textbook is supposed to be free of contradictions). It's true that an encyclopedia isn't a primary or secondary source, but as a reference work I think we should still uphold the same standard of being free of contradictions. I believe this can be done while primarily reporting on what definitions other people use, rather than having to pick and choose. For example, the statements in the continuity article that rely on a different definition of "limit" than the one used in the limit article could be grouped under a section "Alternative definitions", where it can be explained that those definitions/statements rely on a slightly different definition of "limit" (one that also allows limits at non-limit points). That way everything is still being reported on, but we don't have the article containing straight contradictions (through the hyperlink to the "limit" article). Or of course the other way around with the limit article having an "Alternative definitions" section instead.B01010100 (talk) 01:20, 20 August 2013 (UTC)

Newton's notation

I recall Newton using rather than or . Comments? — Arthur Rubin (talk) 23:26, 14 October 2013 (UTC)

This notation is yet used in physics for the derivative with respect to the time. D.Lazard (talk) 01:37, 15 October 2013 (UTC)

Definition of the derivative

Many readers of this article either come from another article through a wikilink of simply need to be recalled of the exact definition. The present state of the article is highly confusing for them: The definition of the derivative appear only at line 56 for the hypothesis (the domain of the function must contain an open interval containing the point of derivation) and at line 61 for the definition itself. Other readers may come to this article to understand some related terminology, such "derivative with respect to a variable" which is used in the lead, and apparently not defined in this article; presently the article is of few help for them. The definition of the derivative is preceded by explanations which, before the definition, may be useful only for people using WP as a textbook (but Wikipedia is not a textbook) and are confusing for the others. For these reasons I'll add immediately after the lead a section "Definition and terminology". D.Lazard (talk) 16:51, 2 December 2013 (UTC)

I disagree with your view of the article. I do not think that many readers of this article come here to recall the exact definition. Instead I think they come here because they do not understand the concept of the derivative. This is why the article starts slowly, building up motivation for the definition, and treating carefully the difference between the derivative at a point and the derivative function.
Because of this I've removed your new section. I'm still open to improvements in the article, but I don't think the new section is a good idea. Ozob (talk) 02:42, 3 December 2013 (UTC)
I disagree with your assertion that the article starts slowly: To understand correctly the first sentences, one has to understand what is the graph of a function, a tangent, the slope of a curve, and other geometrical notions that, although related to the subject of the article are not required to understand what is a derivative, and are of no help to understand, for example, that the velocity is the derivative of the position. Moreover, the article appears to be based on the strange idea that, understanding a notion is easier if the definition is explained before to be given. My opinion is that understanding a definition is impossible if this definition is not given before the explanation. This article is also based on another strange idea that not using the common terminology helps understanding (replacing "limit" by "limiting value", for example). As it is presently, the article is, maybe, useful as auxiliary textbook for US students of elementary courses of mathematics, but it is confusing for all other readers, and in particular for all the readers that are looking for accurate information.
On the other hand, in the section that you have removed, the only technical notions that appear are the notion of real-valued function of a real variable and of limit. As no correct definition of the derivative may exist without these notions, it is definitely impossible to understand the concept of derivative without understanding these two notions. Therefore the removed section starts as slowly as reasonably possible.
Another remark: The heading of the sections are confusing. The subject of section "Differentiation and the derivative" (without its subsections) is "Geometrical interpretation" and the subject of the subsection "Rigourous definition" is "Geometric interpretation of the definition". It cannot be a rigourous definition (what is a non-rigourous definition?) because of the number of geometric terms that do not appear in any correct definition (graph, tangent line, secant line, slope).
D.Lazard (talk) 16:17, 3 December 2013 (UTC)
I think that the notions of graph and slope will be familiar to this article's audience. The audience may be familiar with secant and tangent lines, but maybe not. I would expect them to be least familiar with limits. I might be wrong. My view is limited mostly to America and American education, and it's colored by my own preference for geometry. I do agree that the geometric viewpoint taken in the article will not be much help to readers who are more inclined to, say, physics. Newton's motivation was physical, so perhaps the article should discuss some physics.
However, I think our biggest disagreement is about the purpose of the article. As I see it, you view the article as a reference for information about the derivative. Please correct me if I'm wrong on this, but I think you are making the implicit assumption that the reader has some understanding of what the derivative is good for. My point of view is that the article should not make that assumption. The article should instead assume that the reader does not know what a derivative is and does not know why a derivative is useful. This is why the article builds slowly to the definition: It is attempting to explain to the reader what kind of information is stored in derivative and why the definition of the derivative is the right one to capture that information. It is probable that the article could be better at this. It is possible that rearranging the article so that the definition came first and the justification came later would be an improvement, but I'm doubtful.
I am not sure that limits are necessary to define the derivative of a real function of a real variable. For instance, the Radon–Nikodym theorem does not need limits, only open sets, and it proves the existence of derivatives. Or one could take a Zariski tangent space approach as follows: Consider the ring C0(R, R) of continuous real-valued functions on R. Each point a of R determines a maximal ideal ma, and the derivative of f at a is the class of ff(a) in ma / ma2. There's an obvious one-dimensional subspace of this vector space, namely the span of any non-constant affine function; functions with classical derivatives are those that determine elements of this subspace. (The vector space isn't one dimensional, because it has classes for functions like the absolute value function. Therefore you get some non-classical derivatives this way. I think that for R the vector space should be two dimensional, one dimension each for the positive and negative directions, and for Rn there should be one basis vector for each element of Sn−1. But I don't know how one would prove that.) I don't think these should be covered in the article, but I do think they justify my belief that the intuitive concept of a derivative does not necessarily depend on limits. Ozob (talk) 06:23, 4 December 2013 (UTC)
Radon-Nikodym derivative is not the derivative of a function, but the derivative of a measure. Thus this is not an alternative definition, but a generalization. Your "Zariski tangent space approach" seems WP:OR and appear to be completely buggy: I cannot imagine for ma another definition than the set of functions such that f(a) = 0. Thus ma = ma2. The proper definition of the reals need the notion of limit (or the equivalent notion of least upper bound). The definition of a continuous function involves the notion of limit. Thus in the standard logical model of mathematics (Zermelo-Fraenkel, with axiom of choice) and with the standard definition of of the reals as the smallest complete ordered field containing the rationals, there is a unique definition of the derivative. It may be possible to define it without using limits, but limits must be replaced by another notion which is not really simpler (least upper bound or shadow of a non standard real number). In any case mentioning such approaches in the beginning of the article is confusing for most audience. This, and WP:DUE are the obvious reasons to remove the mention of generalizations and other approaches of the derivatives in the beginning of the article. I have already removed them and I will do it again. D.Lazard (talk) 08:58, 4 December 2013 (UTC)
I agree that all alternatives to limits require concepts that are at least as complicated. I am coming around to your viewpoint that the beginning of the article should discuss the standard definition of the derivative, and other definitions, if they are mentioned at all, should be near the end.
Since functions determine measures, the Radon–Nikodym derivative of a measure determines the Radon–Nikodym derivative of a function. While it is strictly more general, there are no difference quotients involved. One surely needs limits of sequences somewhere in that, but that's a different concept than the limit of a function. It is therefore an approach to derivatives which does not involve limits of functions. Since the Radon–Nikodym theorem is important, I have added a link to it in the "See also" section.
Yes, the Zariski tangent space approach is OR. I hope I did not give any other impression. I am not proposing that we include it in the article. But I do not see why it should be true that ma = ma2. Certainly this is false for a polynomial ring (though we are not in that situation). I think it that on Rn there should be an explicit isomorphism C0(Sn−1, R) → ma / ma2 defined by (in the case a = 0) mapping f to the function that sends x to ||x||f(x/||x||) (i.e., extending f by homogeneity). I am not sure how to prove or disprove this because I don't see any way to find elements in ma2. Ozob (talk) 16:28, 4 December 2013 (UTC)
For elements of , if , then and are in as well, so , so . Remarkably, this approach is not fixed by going up to (despite the fact that this would be begging the question anyway). In that case is much too large (because of the failure of Hadamard's lemma), and you can conclude that a function in is in the congruence class modulo of an affine function for each in some open set only if the function actually is itself affine. (There is a way out of this by replacing by its closure in a natural topology on .) Sławomir Biały (talk) 13:34, 7 December 2013 (UTC)
Beautiful! I'm amazed. Do you have a reference for your statements about ? Ozob (talk) 16:49, 7 December 2013 (UTC)
I don't remember where I saw this. It's possibly an exercise in one of Karl Stromberg's textbooks on analysis, but I don't have a copy handy. Sławomir Biały (talk) 15:44, 10 December 2013 (UTC)

Structure of the article

The main question of the previous thread has been discussed by only two editors, Ozob and myself. This question is about the place in the article of an accurate definition of the derivative. Such a definition does not appear clearly in the present state of the article. I have written a new section including the definition and also the most common jargon (also lacking) here. I have placed this new section at the beginning, because it is always better to define things before talking about them. I have been reverted, and I have not reverted the revert for avoiding an edit war.

Thinking about a solution that may satisfies everybody, it appears to me that the elementary part of this article should have three parts:

Surprisingly, these three parts are almost independent: the formal definition is based on a formal definition of "limit", while the explanation uses (and explains) the intuitive notion of limit (it must be said somewhere and clearly that the formal definition is exactly the formalization of the intuitive notion). The derivation algorithm uses the formal definition only for establishing, once for all, the basic formulas, and does not imply to understand the notion (computer algebra systems do not understand anything :-) Despite this relative independence, my opinion is that the formal definition should appear first, because it is not mathematically correct to develop reasoning about a topic which is not defined. However the formal definition must be preceded by a caveat saying that this formal definition is rarely used in practice, that the next sections do not use it explicitly, and that skipping this section is not a problem.

For these reasons, I'll reinsert the reverted section, with a caveat added, and then (but probably not the same day) edit the remainder of the article to make its structure clearer. D.Lazard (talk) 14:01, 9 December 2013 (UTC)

I haven't followed all the details of the debate about the definition, but at any rate I find your comment about xx puzzling. Why does one need functions of two variables to differentiate this? Apparently chain rule is enough. Tkuvho (talk) 14:23, 9 December 2013 (UTC)
There is a part which the article needs but does not have, and that is history. An encyclopedic article about the derivative is incomplete if it does not have any history. I am not qualified to write such a section. However, I think that if the article had a well-written history section, it would remove the need for an informal section on motivation.
I also think the single variable case should be treated totally separately from the multivariate case. I have never encountered a situation where I would use the multivariate chain rule to compute a single variable derivative, but I have no experience implementing computer algebra systems. I think I understand the xx example as follows: Computer algebra systems do not understand how to take the derivative of uv; but they can be given the rule uv = evlog u, and they can be taught how take the derivative of evlog u using the single variable chain rule (for functions of the form ef(u, v)) and the multivariate chain rule. Is that accurate? Ozob (talk) 14:57, 9 December 2013 (UTC)
(Edit conflict) Ooops. You are right, the generalized chain rule is needed only if there is, among the basic functions, a bivariate function, which cannot be reduced to univariate functions, but whose partial derivatives are known. This may occur with special functions), but this is out of the scope of this article. Thus I agree with the second part of Ozob's post. D.Lazard (talk) 15:20, 9 December 2013 (UTC)
About the history, I am also not qualified to write it, and I agree that it is needed. However, explanation section is, more of less, the description on how Leibniz and Newton came to the notion of derivative. Moreover it seems that there are only two important dates in this history: invention by Leibniz and Newton, and formalization by Weierstrass. It therefore possible that historical comments in the text could be more convenient that a separate section. D.Lazard (talk) 15:20, 9 December 2013 (UTC)
I think the history of multivariable derivatives might be more complicated. For example, I don't know who discovered the total derivative. Maybe it was Newton or Leibniz, but maybe not. It might have been Lagrange or Euler, or it might have been much later. Ozob (talk) 02:29, 10 December 2013 (UTC)
I am also confused about the basis for the discussion of the first bullet point. Does a rigorous definition of the derivative (along with a motivation) not already appear in the Rigorous definition section? Is the terminology not already covered in the Notation for differentiation section? What additions are exactly being proposed? Sławomir Biały (talk) 14:52, 10 December 2013 (UTC)
After cleaning out some of the more glaring issues, I think I can now see more clearly what the issues Professor Lazard raises are. My opinion is that some Introduction section should be split out of the current Differentiation and the derivative section, probably with some editing. This section should include a discussion of the slope of a line, the secant line to a function, culminating in the limit definition (though just an informal treatment of it). Then, the next section (say Definition and notation) can include a brief formal definition of the derivative, a summary of the most popular notations—probably just the Leibniz and Lagrange notations, with a link to the main Notation for differentiation article (and the existing two notations section of this article should be removed). This would look something along the lines of what Lazard originally added to the article. I'm less clear on what is to be done with The derivative as a function, Higher derivatives, and Inflection points, but I would suggest for now grouping these under a top level The derivative as a function heading. Sławomir Biały (talk) 15:42, 10 December 2013 (UTC)

Initial sentence.

"The derivative is a fundamental tool of calculus for studying the behavior of functions of a real variable." This doesn't explicitly define what a derivative is. The definition should be something alone the lines of "A derivative of a function is the instantaneous rate of change of the function with respect to its variable(s)JDiala (talk) 19:22, 15 December 2013 (UTC)".