Talk:Product rule

Proof

Hi,

the proof given for the product rule is a little hard to follow. What about rewriteing it, using the x -> x0 notation? And explaining the steps in more depth (I've had a little difficulty proving the tricky part to myself, when g(x)h(x) - g(x0)h(x0) becomes g(x) (h(x) - h(x0)) + h(x) ( g(x) - g(x0)).) PAStheLoD 18:49, 9 June 2006 (UTC)

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Thanks, Anome. But if we have separate "formal" proofs that:

a) As δx -> 0, δy/δx -> dy/dx (what I have called "differentiation from first principles")
b) The limit of a product is the product of the limits, and vice versa (what I have called the "product rule for limits")

then does that make the proof count as a formal one? Kidburla2002.

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The justification is informal, because it uses infinitesimals in an informal way: it works under most reasonable assumptions, but what about unreasonable ones? To do it formally, you can either use limits formally (with δ - ε type stuff), or a formalised system of infinitesimals. The Anome

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I changed the formal proof; I agree that it was hard to follow. I think it's much easier to prove this as δx -> 0 rather than as x -> x₀. It also fits better with the initial "intuitive" proof, so it makes it clearer how make an intuitive proof rigorous. I tried to be fairly clear (without being pedantic) about definitions, hypotheses, conclusions, and where we use properties of limits. I also was careful to write it in a way that kept f and g in the same order, so that the proof will "work" in the non-commutative case. — Preceding unsigned comment added by Natkuhn (talk • contribs) 00:36, 26 December 2012 (UTC)

Disambiguation

Why was this moved? What is going at Product rule? -- Tarquin

I am not aware of any other product rule, so a disambiguation like product rule (calculus) is unnecessary and I will move back to product rule. AxelBoldt 19:54 18 Jun 2003 (UTC)

The Product Rule is a proper name Pizza Puzzle

There's plenty of other identities which can be called product rules. There must be 15 of them involving curls and divs and grads and what not. But I'm an opponent of pre-emptive disambiguation, so I say leave it unless there's a real need to move it. Certainly you can call this one the product rule, and get away with it. -- Tim Starling 14:24 19 Jun 2003 (UTC)

There is another "product rule" that is often used in combinatorics. (see Rule of product) which (i think) would be better listed as "product rule" than "rule of product". --Hughitt1 23:09, 3 April 2006 (UTC)

"Informal justification" section

I took away the "informal justification" section, for the following reasons:

1. It is virtually identical in essence to the argument using differentials at the beginning of the article. The steps are almost identical, just put "delta-u" instead of "du", or vice versa.
2. There is already a rigorous proof given. In other words, there is already a "formal" derivation (using differentials), and a rigorous proof (using difference quotients, so unless this is really a new idea (and it isn't, see #1), why is it necessary?

Revolver 03:21, 20 Feb 2004 (UTC)

???

This can also be seen as a barber shop analogy. For example, in the above example, u stands at one side while v takes the haircut.

Is this just silly vandalism, or am I really missing something??? Revolver 16:11, 30 September 2005 (UTC)

I couldn't understand this either and have removed it. Eric119 08:34, 16 October 2005 (UTC)

Leibniz and intuition

The argument of Leibniz can be adapted to provide better intuition than more common proofs.

Let ρ be an infinitesimal, and write g(x+ρ) as g(x)+g′(x)ρ+O(ρ²); likewise for f(x+ρ). Then (f·g)(x+ρ) will be

${\begin{aligned}(f\cdot g)(x+\rho )&{}=f(x+\rho )\cdot g(x+\rho )\\&{}=\left(f(x)+f'(x)\rho +O(\rho ^{2})\right)\cdot \left(g(x)+g'(x)\rho +O(\rho ^{2})\right)\\&{}=f(x)g(x)+f'(x)g(x)\rho +f(x)g'(x)\rho +O(\rho ^{2})\end{aligned}}$

When we take the derivative we subtract off the f(x)g(x) term, divide by ρ, and discard any remaining terms containing ρ — which will be the O(ρ²) bit. In other words, the product rule is merely the distributive law followed by the discarding of higher order terms. This also quickly explains the result for a multiple product; any product term with two or more derivatives will be higher order, thus discarded.

The derivative here is, of course, a non-standard analysis version, of the form

$f'(x)=\mathrm {standard} \left({\frac {f(x+\rho )-f(x)}{\rho }}\right),$

where ρ is an infinitesimal.

Sometimes this idea is conveyed with a picture of areas.

░░░░░░░░░░░░█
▒▒▒▒▒▒▒▒▒▒▒▒░
▒▒▒▒▒▒▒▒▒▒▒▒░
▒▒▒▒▒▒▒▒▒▒▒▒░
▒▒▒▒▒▒▒▒▒▒▒▒░
▒▒▒▒▒▒▒▒▒▒▒▒░
▒▒▒▒▒▒▒▒▒▒▒▒░

Here the top right corner corresponds to the higher order terms in ρ. --KSmrq^T 04:26, 28 January 2007 (UTC)

multiplication signs, please

Formulas like

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

${d \over dx}(uv)=u{dv \over dx}+v{du \over dx}$

are hard to decipher for the beginner.

Multiplication signs

$(f\cdot g)'=f'\cdot g+f\cdot g'\,$

${d \over dx}(u\cdot v)=u\cdot {dv \over dx}+v\cdot {du \over dx}$

make it easier.

Bo Jacoby (talk) 21:43, 11 June 2008 (UTC).

For vector functions

I'm new (sorry about sloppy/lack of code) and not a math-whiz (sorry if this is irrelevant), but in reading this article I thought it might be worth noting in the section "For vector functions" the difference between dot and cross products in this circumstance. Or perhaps just denoting the equation with the appropriate multiplication sign as noted above would suffice.

... that is :

$(f\times g)'=f'\times g+f\times g'\,$ and not

$(f\times g)'=f'\times g+g'\times f\,$

7yl4r (talk) 23:17, 24 February 2009 (UTC)

Associativity ?

I think it is very unfortunate that associativity is assumed throughout this article without ever mentioning it, then, under Generalisations suddenly the sequence of multiplicands is observed. IMHO the product rule should NEVER be stated

${\dfrac {d}{dx}}(u\cdot v)=u\cdot {\dfrac {dv}{dx}}+v\cdot {\dfrac {du}{dx}}$

but only

${\dfrac {d}{dx}}(u\cdot v)=u\cdot {\dfrac {dv}{dx}}+{\dfrac {du}{dx}}\cdot v$

or with brackets, where necessary

${\dfrac {d}{dx}}(u\cdot v)=u\cdot {\dfrac {d}{dx}}v+({\dfrac {d}{dx}}u)\cdot v$

This would be appreciated by everyone who works with operators (quantum mechanics), matrices, or just wants to apply the product rule to vector products (as 7yl4r found out already). In this respect, unfortunately, many of the math lines need reworking - anyone? 213.68.42.99 (talk) 13:15, 19 January 2010 (UTC)

Did you mean commutativity rather than associativity? Commutativity of multiplication says ab = ba. Associativity is something else. Michael Hardy (talk) 14:21, 19 January 2010 (UTC)

The problem with stating it right is that students will go and forget that they aren't supposed to differentiate the v, very silly I know. Also it makes the two bits look different even though multiplication is commutative for what they're doing. Dmcq (talk) 16:34, 19 January 2010 (UTC)

source for barrow

See "The early mathematical manuscripts of Leibniz", Gottfried Wilhelm Leibniz, translated by J. M. Child; page 29, footnote 58. Tkuvho (talk) 15:44, 18 October 2010 (UTC)

that citation you gave said nothing of the sort, perhaps you could point to the version of the book as well or indicate the place better? I'm not doubting Barrow made a geometrical analogue but lots of people did geometrical bits of calculus before it was formalised in an algebraic way. Also I would have thought more should be put into Barrow's article about his geometric lectures before branching out as it were? Dmcq (talk) 16:11, 18 October 2010 (UTC)

The 2008 edition is online. I am not sure what you mean above. You were not able to find footnote 58 on pages 28-29? Tkuvho (talk) 17:23, 18 October 2010 (UTC)

I see what has happened. Okay I've read it and in it Leibniz is accused of plagiarism as far as I can see. Even though Leibniz quite evidently got it wrong in the first instance. That is so self-evidently silly I don't know why Child wrote it. However I can see the attribution of the product rule to Barrow. I still believe you would be much better off adding a bit about Barrow's geometrical lectures in his article for the reasons I gave above, Child wrote a book about them. Dmcq (talk) 18:22, 18 October 2010 (UTC)

I read Child somewhat differently. My guess is that Barrow had it first, Leibniz discovered it independently, and Newton I don't know. You seem to know more about Barrow's geometrical lectures than I do; why don't you give it a try? I can participate if you wish. Tkuvho (talk) 20:08, 18 October 2010 (UTC)

Leibniz did not make this error

The claim that "Leibniz himself made this error initially" is absurd, though it has been recycled in some publications more interested in amusing the students than historical accuracy. Leibniz specifically says that he would like to determine whether the rule d(xy)=dxdy applies, makes a few calculations, and a few lines later rejects this. This was dealt in detail in Margaret Baron's book (last chapter). Tkuvho (talk) 15:54, 18 October 2010 (UTC)

Well stick the citation in saying it is disputed. What's there is from the same J M Child book you referenced above to support Barrow. Dmcq (talk) 16:17, 18 October 2010 (UTC)

I can't follow your grammar. The source for the product rule is Baron, not Child. Tkuvho (talk) 17:32, 18 October 2010 (UTC)

I really wish you would put proper citations into the article. Just copy another citation to see how to do them. If you can give details on the talk page then you can do enough for that. You've put in an addition but left the original citation that contradicts what you said. That citation references Michelle Cirillo who used Child's book and another one by Antonella Cupillari

As to Baron I just had a look and all it seems to say about that is that he 'eventually' worked it out. Dmcq (talk) 18:00, 18 October 2010 (UTC)

I see. I might have misread Baron. If so, my apologies. Do you have the impression Baron would agree with the claim that Leibniz originally made a mistake? Tkuvho (talk) 20:09, 18 October 2010 (UTC)

Note that Leibniz clearly states that he wishes to examine whether d(xy)=dxdy, or not. He proceeds to examine it, and comes to a different conclusion. This much emerges clearly from Baron's documentation. By no stretch of the imagination can this be described as an "error" on Leibniz's part. Does Cupillari actually examine the evidence? Cirillo certainly does not. Tkuvho (talk) 20:59, 18 October 2010 (UTC)

Well I think apocryphal is too strong when he spent a couple of pages on the idea and only wrote the correct result 10 days later. See Church (2008) page 100 on about it. Especially the bit on page 101 where he says it appeared to be proved but was incorrect. Incidentally Child on page 102 seems to attribute the original idea to Fermat and then Barrow and Newton, I'm not sure what that's about. Dmcq (talk) 21:54, 18 October 2010 (UTC)

Let's get back to Barrow later. If Leibniz wrote that "it appeared to be proved but was incorrect", doesn't it show that he never actually claimed d(xy)=dxdy as true? And if the text we are quoting is a running diary where he felt free to improvise over a 10 day period, isn't it ungenerous to claim that Leibniz made an "error" as if there was a mistake in his research paper? Tkuvho (talk) 02:33, 19 October 2010 (UTC)

Well if you can phrase it better please do. However apocryphal refers to stories which didn't happen in real life. He though it might be true and later found out it wasn't. If he thought 7 times 8 might be 54 and later corrected it to 56 I know what I'd call it. Anyway we should be following the sources and not giving our own opinions, if there is dissent about the point then put the citation for the other point of view into the article as well please or else show how the citation that's there is unreliable or plain wrong. As far as I can see it is not plain wrong but a matter of interpretation. Dmcq (talk) 12:27, 19 October 2010 (UTC)

Can you suggest a wording based on your reading of the secondary sources? As far as Cirillo is concerned, I don't think we need to relate to her as she has no pretense of being a historian. In fact we should delete the reference. Tkuvho (talk) 12:53, 19 October 2010 (UTC)

Example

Isn't the law only valid for constant-mass systems?123Mike456Winston789 (talk) 22:05, 19 January 2011 (UTC)

For constant mass you would get F=m(dv/dt), i.e. F=ma. The formulation F=dp/dt is more general. Tkuvho (talk) 22:16, 19 January 2011 (UTC)

${\text{st}}$ notation

I'm not versed in non-standard analysis. What does the ${\text{st}}$ notation in the "Using non-standard analysis" section mean? (I'd look it up in a non-standard analysis article but somehow I'm not comfortable editing a section about which I know very little.) 67.158.43.41 (talk) 05:57, 4 January 2011 (UTC)

st denotes the standard part function. I have added a few words and links to the section to provide some minimal context. It still is rather bare I'm afraid. — Tobias Bergemann (talk) 08:58, 4 January 2011 (UTC)

Thank you, that's all I wanted. The st page is itself bare, but ah well. Maybe one day I'll have to read a book on non-standard analysis. 67.158.43.41 (talk) 13:17, 9 January 2011 (UTC)

What is your background exactly? To get some motivation it may be helpful to start at Ghosts of departed quantities. The reason standard part function is "bare" is because it relies on other pages to explain the background. In an encyclopedia, one can't expect each article to expain every point. Tkuvho (talk) 01:44, 10 January 2011 (UTC)

incorrect calculation in the section "using nonstandard analysis"

The calculation in the section "using nonstandard analysis" is incorrect. Tkuvho (talk) 15:20, 13 February 2011 (UTC)

Explain. At first glance, it looks acceptable, although verifying

$\operatorname {st} {\frac {\mathrm {d} u\cdot \mathrm {d} v}{\mathrm {d} x}}=0$

is not obvious. — Arthur Rubin (talk) 17:18, 13 February 2011 (UTC)

That step is OK, but the calculation confuses the infinitesimal dx, dy and their role in the symbolic notation for the derivative dy/dx. Dropping the mention of "st" in the last line is not correct if dy/dx is literally a ratio, since it may not even be real. Tkuvho (talk) 17:22, 13 February 2011 (UTC)

OK, I can see the problem, related to the confusion of the non-standard analysis expression:

$\operatorname {st} {\frac {\mathrm {d} u}{\mathrm {d} x}}$

and the calculus expression

${\frac {\mathrm {d} u}{\mathrm {d} x}}$

But what to do about it? — Arthur Rubin (talk) 17:25, 13 February 2011 (UTC)

What Keisler and others usually do is use $\Delta x$ for the initial infinitesimal. Once the derivative is defined as the standard part of $\Delta y/\Delta x$, one defines dy by setting dy = f'(x)dx, where $dx=\Delta x$. Tkuvho (talk) 17:29, 13 February 2011 (UTC)

"Common Error" - really?

The article says "It is a common error, when studying calculus, to suppose that the derivative of (uv) equals (u ′)(v ′)". I dispute that. I don't see why it's intuitive (and the article doesn't say), and in my experience most students simply follow whatever recipe they're given. But leaving aside my personal opinion, it's unreferenced. It's been sitting there since the very first version of this article (Nov 2002) when WP was in its infancy,[1] and it's time it was removed. The reference for Leibniz' error can simply be moved to the history section. Adpete (talk) 11:52, 25 February 2013 (UTC)

OK I'm removing it. Even in the unlikely event that it's true, it's not encyclopedic and (to me) is out of place in an otherwise excellent article. Adpete (talk) 06:34, 26 February 2013 (UTC)

Formatting error.

I'm getting the error

"Failed to parse(unknown function '\begin'): {\begin{aligned}d(u\cdot v)&{}=(u+du)\cdot (v+dv)-u\cdot v\\&{}=u\cdot dv+v\cdot du+du\cdot dv.\end{aligned}}" 

in the first line of the discovery section. I don't know enough about wiki formatting math equations to fix this or if this might just be an error with my browser. Someone with better experience of formatting maths expressions should probably have a look at fixing it. — Preceding unsigned comment added by 110.20.69.222 (talk) 07:56, 10 February 2014 (UTC)

Rigorous to Direct?

Two proofs of the Product Rule are provided. As far as I can see they are both rigorous and they are both nicely presented. However, the title of the second proof, `Rigorous Proof', suggests, to me at least, that the proof by factoring is not rigorous. The second proof proceeds directly from the definition of the derivative. I suggest changing the title to `Direct Proof'. Michealefr (talk) 08:24, 13 September 2015 (UTC)

Wikipedia_talk:WikiProject_Mathematics#Article_product_rule

Please take a look at Wikipedia_talk:WikiProject_Mathematics#Article_product_rule. Dennui (talk) 06:19, 23 April 2019 (UTC) [2]

https://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mathematics/Archive/2019/Apr#Article_product_rule Dennui (talk) 13:53, 5 July 2020 (UTC)

Is this limit superfluous?

In the final line of the proof it seems to me that the limit applied to f(x) [i.e., "\lim_{\Delta x\to 0} f(x)"] shouldn't be there. After all, f(x)=f(x) by definition. This is a minor thing but if it's superfluous it makes the proof more complicated than it needs to be. Joe in Australia (talk) 21:39, 27 October 2022 (UTC)

The author of the proof was mechanically distributing the limits over sums and products. Then the limit in question simplifies in the next step. Whether to simplify it a step earlier, during the distribution step, is a matter of taste. For what it's worth, I prefer the current version, over the version that you seem to be proposing. Regards, Mgnbar (talk) 23:21, 27 October 2022 (UTC)

comparison to it's product rule: cross term

the current articles doesn't discuss dudv and why this is not used In differentiable deterministic calculus but is required in stochastic calculus. Ito's product rule is not cited or referenced in article which is therefore missing discussion of important concepts. Rdsk2014 (talk) 08:22, 5 November 2023 (UTC)

It seems that what you call dudv is a differential form of order 2. This has nothing to do with the product rule, and this is described in details in the linked article. Product rule is about differentiation of a product of functions, not about the exterior product of differential forms. D.Lazard (talk) 12:06, 5 November 2023 (UTC)

plausible search not supported

If I google for "product rule for derivatives", this article should be among the first hits presented. As it is, it doesn't appear at all. So, this expression ought to be incorporated into the article in some way, perhaps as part of a list of keywords.

Big-picture takeaway: I guess we can't just blithely assume that a garrulous / comprehensive article on a topic will automatically contain the most plausible / relevant search strings for it. Maybe a deliberate program of inclusion of keywords for every article needs to be undertaken. Kontribuanto (talk) 14:58, 5 March 2024 (UTC)