Talk:Derivative at a point

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

${\frac {df(x)}{dx}}:=\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}}=\lim _{\Delta x\rightarrow 0}{\frac {\Delta f}{\Delta x}}.$

--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)