# Talk:Derivative at a point

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 Definition:  The local rate of change of a function with respect to its argument. [d] [e]
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## Derivative

Peter, could you please explain why you prefer the title "differential quotient"? I haven't studied mathematics in English for some time, but I still feel that "derivative" is the more common name. Formally, the derivative should be the limit of the differential quotient as h approaches zero, but in my mind they are not the same concept. Johan A. Förberg 22:08, 21 January 2011 (UTC)

I see a subtle difference:
• The differential quotient of f at x is the limit of the difference quotients at x (only one particular point considered),
• while the derivative of f is the function with values equal to the differential quotient (the full dominion of the function is considered).
(The redirect is not final, "derivative" should have its own page, as should have "derivation".)
Peter Schmitt 00:54, 22 January 2011 (UTC)
OK, I see your point. But as the article reads now, it only confuses the reader further as to the difference between the derivative and the d.q. Johan A. Förberg 23:34, 22 January 2011 (UTC)
I never met the term "differential quotient". Wikipedia has no such article, and moreover, its search gives no results. Google gives first 5 results that contain in fact only "difference quotient", but result no. 6 (dictionary.com) mentions "differential quotient" as item 6 in "derivative". --Boris Tsirelson 06:34, 23 January 2011 (UTC)
Yes, I was also surprised that it popped up so rarely, but it does so in different places, including research papers.
Could it be a Germanism? The term is very usual in German. I'll try to find out more in the literature -- old and new. This may help to deal with it properly.
From a didactical perspective, it is a rather useful distinction -- e.g., you need a derivative (function) before you can talk about s second derivative.
--Peter Schmitt 10:58, 23 January 2011 (UTC)
I believe the confusion goes back to the Newton-Leibniz controversy. Newton talked about fluents and fluxions and Leibniz about differentials. The continent followed Leibniz (one of the first things I learned in Delft, a continental city, was the word "differential quotient") while England stayed with Newton. In the 19th century the British changed slowly to the Leibniz notation, but they did not adapt his complete terminology. The typical British book by Wittaker-Watson (1902) doesn't use the term "differential quotient", while the modern German DTV-Atlas zur Mathematik gives it. As far as I know there is no clear distinction between derivative and DQ. For what it is worth I (not being a mathematician) would simply write
${\displaystyle {\frac {df(x)}{dx}}:=\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}}=\lim _{\Delta x\rightarrow 0}{\frac {\Delta f}{\Delta x}}.}$
--Paul Wormer 12:58, 23 January 2011 (UTC)

I am not and never have been a professional mathematician, although I use mathematics. It may, indeed, be linguistic. From the 1960s, I still have several American calculus textbooks, and none appear to use the term. While I think I understand Peter's distinction between instantaneous and range, my sense is that is an advanced point.

Renaming articles, I believe, needs some discussion first. Incidentally, have any of you read the New York Times series on popularizing "advanced mathematics" -- advanced to the layman? It has a nice introduction to the idea and history of derivatives, which it compares and contrasts, historically, to integrals. Howard C. Berkowitz 18:36, 23 January 2011 (UTC)

Peter, would you be willing to revert your page move? Johan A. Förberg 21:21, 23 January 2011 (UTC)
German WP: Hierzu dient die Ableitung (auch Differentialquotient genannt). [The derivative (also called differential quotient) serves to this end.] --Paul Wormer 08:45, 24 January 2011 (UTC)
The actual English equivalent of dq is "differential coefficient". Peter Jackson 12:10, 24 January 2011 (UTC)
As far as I can tell (e.g., from the WP article) the differential coefficient is again the derivative (function). As I said previously, I'll try to research this before I form an opinion. --Peter Schmitt 12:22, 24 January 2011 (UTC)
Without researching, it seems to be usual among mathematicians, to say either "the derivative of f at x equals k", or "the derivative of f equals g". In the former case it is a single number, in the latter case a function, but the term "derivative" can serve both. --Boris Tsirelson 16:17, 24 January 2011 (UTC)

[unindent]

I happened to be reading Stephen Hawking's translation of Einstein's 1916 paper and saw the term "differential quotient" for ${\displaystyle \chi ={\frac {d\psi }{ds}}}$. --Paul Wormer 13:21, 26 January 2011 (UTC)

Could he have been just translating literally? Peter Jackson 10:38, 2 February 2011 (UTC)

## "Derivative" as an article title is preferable to "Differential quotient"

In my experience the term "differential quotient" hardly ever comes up, while "derivative" and "differential" occur commonly. I'd say the title of this page is now presenting a secondary, minor usage, at least as far as American and English usage. To be re-directed from "derivative" to "differential quotient" appears to suggest that "differential quotient" is the more common and preferred term, which I'd dispute. John R. Brews 19:12, 22 February 2011 (UTC)

An historical account can be found in Boyer, p. 275 where the term "differential quotient" is described as an invention of Leibniz in a formulation based upon differentials, but subsequently overturned by Cauchy, who introduced the derivative in terms of limits and the term "differential" in terms of the derivative. This source attributes to Cauchy a formal precision previously lacking. In my view this settles the matter that the article should be returned to the title "Derivative", and a Redirect used to send "Differential quotient" to this page.

If more is wanted, this google book search turns up 119,000 results for "differential quotient", of which many are unrelated to derivative. On the other hand, this google book search turns up 1.8 million hits for "derivative". John R. Brews 22:45, 22 February 2011 (UTC)

I now moved the page ("Derivative at a point"). Essentially, it only deals with that case, and the derivatve function deserves its own page, I think.
By the way, because of this difference the Google search is not as overwhelmingly convincing as it may seem at first glance. -Peter Schmitt 14:16, 23 February 2011 (UTC)
The term "derivative at a point" can be seen to be separate from the "derivative function" assembled from all the "derivative-at-a-point" values. However, the distinction is seldom used. This google book search turns up only 2,330 hits for "derivative at a point", showing much less usage than either "derivative" or "differential quotient". Just what is the purpose of separate articles on these two topics? Does it reflect common usage of two quite different ideas with widely varying application? Or, does it only emphasize a distinction that is occasionally useful, but not often found in everyday science or engineering? John R. Brews 20:18, 23 February 2011 (UTC)
In everyday science or engineering probably not, indeed. But still, mathematically, differentiability at a given point has no logical relation to differentiability at any other point. It is possible to define the derivative (function) globally, without first treating each point separately; however, this way probably is seldom used in teaching for mathematicians, and never - for others. That is, the derivative-function is usually treated as just the collection of all derivatives-at-points. --Boris Tsirelson 21:31, 23 February 2011 (UTC)
Hi Boris: So, to try to connect the dots here for all but a few special readers, being directed from "derivative" to either an article titled "derivative at a point" or one called "derivative function" would be neither here nor there? If both these articles were written, it would make no never-mind to even the average scientist which article they read: they'd get what was for them the same info in both articles apart from some delicate questions they probably never would think to ask? John R. Brews 23:17, 23 February 2011 (UTC)
Boris already answered this. Whether you call it derivative or differential quotient, differentiability is a local phenomenon. It may exist at a few isolated points only. And only in special cases it can be used to define a derivative function. (Even in this case the locally defined limits are used.) That in science functions are usually assumed to be (almost) everywhere differentiable does not change the logical dependence. This should be of interest to all scientists. (And the number of Google hits has no bearing here.)
When all articles will be written, derivative will have to be a disambiguation page (not only for mathematics! There are non-mathematical meanings, too). We should have derivative of a function (and derived set) and derivation (or similar) that properly reference each other. (Maybe some other articles, too.)
--Peter Schmitt 23:59, 23 February 2011 (UTC)

OK: The vision is a very deep CZ with a lot of very detailed explanations and what some might call nuanced or specialized or even hair-splitting articles. That is not how it is at the moment: the vast majority of articles I've looked at stick to the qualitative and make no attempt at what might be called a "treatise" level of sophistication. What is the vision of CZ as you see it? And is it possible that a staged evolution to greater complexity, beginning with rather broad treatments and using them as background introductions later on, would be a suitable route to take, given that there are a million holes in CZ coverage at the moment, never mind this kind of detail? John R. Brews 00:21, 24 February 2011 (UTC)

Yes, currently CZ has more gaps and poor articles than good ones. And yes, I hope that some time in the future there will be a lot of "deep and detailed" explanations. But what we are talking about here is neither deep nor hair-splitting: Every textbook has to begin with differentiation at a point, perhaps in form of a tangent. First it has to define f′(x0) and to follow with f′ later. --Peter Schmitt 02:13, 24 February 2011 (UTC)
All this is another manifestation of a general fact: no text about some mathematics (and not only math, I guess) can satisfy mathematicians, physicists, engineers, economists etc. simultaneously. This is why in the real world (outside wikis) we observe a lot of textbooks (on the same topic) intended for different audience. I believe that it is a fundamental error of both Wikipedia and Citizendium, to try to satisfy everyone by a single text. It leads all the time to conflicts, inevitably. In contrast, "we allow multiple articles, written from different approaches, on individual topics" [1]. --Boris Tsirelson 14:12, 24 February 2011 (UTC)
You are right, Boris, some/many topics can profit from multiple articles. But the concept of a single article is older, it comes from printed encyclopedias, I would guess. I hope that CZ will last and develop and flourish, and in the end will have such articles. But now. for most topics, we do not have even one good article, let alone several. (This needs authors, and KI has even less ...)
A good example for parallel articles would be vector and tensor analysis. The derivative, however, is not: I don't think that physicists introduce it differently -- the difference comes later, when the average physicist assumes functions to be differentiable (or analytic).
In contrast to textbooks, however, CZ should put parallel expositions in relation to each other, and help to bridge the gap between different cultures instead of supporting it. For instance, most physicists will not need to know non-differentiable functions, but they should know that they exist.
--Peter Schmitt 23:27, 24 February 2011 (UTC)
John, even if there were indeed many readers who do not need/appreciate/want "sophisticated" information there are also other reader (mathematics students, for instance) for whom this is important (and fundamental) information. --Peter Schmitt 00:25, 25 February 2011 (UTC)