Talk:Differential calculus

Introduction

This article could use a smoother, less-technical intro. --DanielCD 21:52, 21 October 2005 (UTC)

Merge

This article used to redirect to, firstly Calculus and then Derivative, before it was started again in its present unreadable form. The section "Differential calculus" in the Calculus article, mentions main article as Derivative, which is thoroughly readable. I suggest the two articles be merged. However, I don't know which name would be better to keep - possibly "Differential calculus" seeing as this is the mathematical process? Cormaggio ^@ 14:47, 11 November 2005 (UTC)

This page is a mess, salvage what you can and mix with something else Signed_in 10:04, 18 November 2005 (GMT)

New version

As discussed on Talk:Derivative, Differential calculus is being turned into the lead article for (surprise!) Category:Differential calculus. It will not focus on the derivative operator (as the old differential calculus article did), but instead on history and applications. The present version is mostly grabbed from the present derivative article, plus some additions (mostly the sketch of the derivative operator). 141.211.62.20 03:28, 28 July 2007 (UTC)

Oh dear

This has a textbook tone! Needs to be fixed :) Uxorion 23:11, 10 August 2007 (UTC)

Go for it.  :-) 141.211.120.81 23:41, 10 August 2007 (UTC)

WAY WAY to complicated, less meat and more show offy symbols and explinations

Shall i ad a parra on the rules in differentiation. Eg product rule, quotient rule, chain rule etc as well as differentiation of trignometric values? the article basically talks about the history of differentaition-(yawn) and also how y' (y dash) can be written differently. WHERE IS THE MEAT!! Forgot to sigh, addy g in da houseAddy-g-indahouse 11:04, 28 August 2007 (UTC)

Scrap that, the rules are there, but they look way way too complicated, for a topic that is merely pre calculus. addy gAddy-g-indahouse 22:59, 28 August 2007 (UTC)

the explanations are WAY WAY to complicated, meaning anyone who wants to learn how to differentiate will look at it wil a expression similar to being clubbed over the head. I have many textbooks that have the rules written in this way, however it can be simplfied, without all the f'(gh'-1 etc.

simple eg product r(ule (x 1)^2(2x 1)^5

Let u=(x 1)^2 v=(2x 1)^5

dy/dx= u'v v'u

simple as that, and without all the (f'(g'h)-1)gh-1

As per proofs, such proofs can be proved such that a newbie to calculus can just look at it and understand it, without calling in the aid of a rocket scientist.

regards addy g in da houseAddy-g-indahouse 11:04, 28 August 2007 (UTC)

I have no idea what you're talking about. This article does not mention the product rule, because it's an overview of the whole field of differential calculus and the product rule is only a small part of it. The product rule is briefly mentioned in derivative as

$(fg)'=f'g fg'\,$

which seems pretty much what you want, and more fully in product rule. -- Jitse Niesen (talk) 01:34, 29 August 2007 (UTC)

Functions that are not lines

This edit ought to be undone. There has been discussion about this on the reference desk. I am not yet convinced that an infinite set can be bigger than another infinite set, plus that is not what the sentence intends to mean in the first place, as User:KSmrq pointed out. A.Z. 06:38, 30 September 2007 (UTC)

Fair enough. I dislike the "many functions of practical interest" though, because it is unnecessarily vague, and the expression "line function" is not used as far as I know. So I used another formulation which resolves the issue of what "most functions" means.

I have to think a bit about the next sentence, which reads "The derivative of f at the point x is the best linear approximation, or linearization, of f near the point x." I think this is an unfortunate way to put it. If linearization of f is the function $x\mapsto ax b$ then the derivative is a; these are not the same. -- Jitse Niesen (talk) 12:59, 30 September 2007 (UTC)

The problem seems to be that the derivative, when formulated abstractly, already encodes the value of the function: Given f : R → R, the derivative of f at a is a linear map f'(a) : T_aR → T_f(a)R. So by the very choice of domain and range, the derivative includes the point at which it's taken and the value of the function at that point. From this abstract perspective, it's entirely appropriate to say that the derivative is the linearization of f at a; but unless you introduce tangent spaces and begin talking in high-powered language, that statement is out of reach, and the linearization of f is the linear function f(a) f'(a)(x-a) instead.

I've replaced the previous language with something that's a little looser to start with ("best possible approximation to the idea of the slope") and is completely precise at the end ("Together with the value of f at x, the derivative of f determines the best linear approximation"). I think this is better, but feel free to edit it as you like. 141.211.62.20 01:26, 18 October 2007 (UTC)

Circular proof

At the beginning of this article is a statement that for a straight ine, delta y/delta x = m, the slope of the line. The "proof" of this statement casnnot be derived without assuming the statement is true. The same logical error (the identical text)appears in the article "differential equation". My guess is that something was lost while changing the article from earlier versions.

Something that seems to be ignored in mathematical articles is that those who look them up, are not familiar with their contents. Those are the people we are trying to help.Don Seib 20:18, 6 October 2009 (UTC) —Preceding unsigned comment added by SEIBasaurus (talk • contribs)

Ian Pearce

The reference to Ian Pearce should be removed. Tkuvho (talk) 10:29, 17 May 2011 (UTC)

Misuse of sources

This article has been edited by a user who is known to have misused sources to unduly promote certain views (see WP:Jagged 85 cleanup). Examination of the sources used by this editor often reveals that the sources have been selectively interpreted or blatantly misrepresented, going beyond any reasonable interpretation of the authors' intent.

Please help by viewing the entry for this article shown at the page, and check the edits to ensure that any claims are valid, and that any references do in fact verify what is claimed.

I searched the page history, and found 3 edits by Jagged 85 (for example, see this edits). Tobby72 (talk) 20:22, 20 January 2012 (UTC)

tangent

I don't understand the ill. at the top of the article. Why is that particular tangent the derivative? There are lots of other possible tangents to the curve. Why is that one "privileged"? 211.225.33.104 (talk) 16:01, 4 May 2014 (UTC)

The slope of that tangent line is the derivative of the function at the marked point. The function has many, many derivatives, one for each point. The only reason for singling out this particular point, this particular derivative, and this particular tangent is because it makes a nice illustration. Ozob (talk) 16:08, 4 May 2014 (UTC)

Proposed merge with Constant coefficients

This is a stub about differential calculus that does not need its own article and is on the abandoned articles page. Ethanpet113 (talk) 07:34, 31 October 2018 (UTC)

We have also Constant coefficient that redirects to Ordinary differential equation. This target and the proposed merged article do not contain any explicit definition of the concept. As "constant coefficients" is generally used for linear differential equations only, and as Linear differential equation contains a definition and several sections about equations with constant coefficients, I'll be bold and redirect Constant coefficient and Constant coefficients to this article. D.Lazard (talk) 09:37, 31 October 2018 (UTC)

Derivative—Simplification of the definition

There is an issue with the section titled 'Derivative', a section I edited heavily. In the section, I use the word 'tangent line' freely, and assume that the reader will have some geometric interpretation of what the tangent line is. However, strictly speaking, the tangent line to $x=a$ is the line with a gradient of $f'(a)$that goes through $(a,f(a))$. This means that it is circular to write that 'the derivative of a function is defined as the slope of the tangent line'. Really, the fundamental object is the derivative, defined as the limit of the difference quotient; this object defines what we mean by tangent line. However, I didn't want to bombard a newcomer to calculus with formalism, and so I did not mention this in the section. Is there a way of improving the accuracy of this section without harming its readability?

Tusi

We have:

He proved, for example, that the maximum of the cubic ax2 – x3 occurs when x = 2a/3, and concluded therefrom that the equation ax2 — x3 = c has exactly one positive solution when c = 4a3/27, and two positive solutions whenever 0 < c < 4a3/27.[9] The historian of science, Roshdi Rashed,[10] has argued that al-Tūsī must have used the derivative of the cubic to obtain this result.

But "proved" doesn't make sense: if we don't know how he obtained the result, we don't know he had a proof. Presumably it was just a "this is the answer" type of thing William M. Connolley (talk) 09:44, 21 October 2021 (UTC)

I have replaced "proved" by "obtained". I have also avoided the minus sign in the equation, as it seems an anachronism (AFIK, negative numbers were unknown at Tusi's time). Also, I have added "(for positive x)" after "maximum", as, otherwise, the function does not have a (global) maximum. D.Lazard (talk) 11:13, 21 October 2021 (UTC)

Discussion

I have started a discussion of possible interest to followers of this page at Wikipedia talk:WikiProject Mathematics#Articles on "differential calculus" and "integral calculus". --Trovatore (talk) 19:10, 27 December 2021 (UTC)

Aryabhatta

i have removed the claim that aryabhata knew about early differential calculus since the reference doesn't mention anything about calculus in aryabhata work David dclork li (talk) 07:59, 22 April 2023 (UTC)