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2857:. 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,
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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.
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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
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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
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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
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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
2006:
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
1973:
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
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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
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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
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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
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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
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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
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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.
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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
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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 Ξ΄ - Ξ΅ type stuff), or a formalised system of infinitesimals.
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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.
832:(Ο) 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.
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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.
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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.
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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?
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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.
802:{\displaystyle {\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}}}
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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.
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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
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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
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In the final line of the proof it seems to me that the limit applied to 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.
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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.
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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)).)
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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.)
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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
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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.
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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.
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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?
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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.
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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.
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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.
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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.
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See "The early mathematical manuscripts of
Leibniz", Gottfried Wilhelm Leibniz, translated by J. M. Child; page 29, footnote 58.
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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.
1938:, very silly I know. Also it makes the two bits look different even though multiplication is commutative for what they're doing.
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b) The limit of a product is the product of the limits, and vice versa (what I have called the "product rule for limits")
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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?
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calculations, and a few lines later rejects this. This was dealt in detail in
Margaret Baron's book (last chapter).
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The problem with stating it right is that students will go and forget that they aren't supposed to differentiate the
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have thought more should be put into Barrow's article about his geometric lectures before branching out as it were?
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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 Ξ΄x -: -->
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the proof given for the product rule is a little hard to follow. What about rewriteing it, using the x -: -->
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and it's time it was removed. The reference for
Leibniz' error can simply be moved to the history section.
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As to Baron I just had a look and all it seems to say about that is that he 'eventually' worked it out.
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attribute the original idea to Fermat and then Barrow and Newton, I'm not sure what that's about.
967:βββββββββββββ βββββββββββββ βββββββββββββ βββββββββββββ βββββββββββββ βββββββββββββ βββββββββββββ
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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For constant mass you would get F=m(dv/dt), i.e. F=ma. The formulation F=dp/dt is more general.
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OK, I can see the problem, related to the confusion of the non-standard analysis expression:
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Did you mean commutativity rather than associativity? Commutativity of multiplication says
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The argument of Leibniz can be adapted to provide better intuition than more common proofs.
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for the initial infinitesimal. Once the derivative is defined as the standard part of
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What is your background exactly? To get some motivation it may be helpful to start at
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I can't follow your grammar. The source for the product rule is Baron, not Child.
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definition of the derivative. I suggest changing the title to `Direct Proof'.
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that Leibniz made an "error" as if there was a mistake in his research paper?
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I took away the "informal justification" section, for the following reasons:
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The calculation in the section "using nonstandard analysis" is incorrect.
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Here the top right corner corresponds to the higher order terms in Ο. --
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Cupillari actually examine the evidence? Cirillo certainly does not.
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with the appropriate multiplication sign as noted above would suffice.
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Explain. At first glance, it looks acceptable, although verifying
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I am not aware of any other product rule, so a disambiguation like
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Is this just silly vandalism, or am I really missing something???
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dy/dx (what I have called "differentiation from first principles")
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incorrect calculation in the section "using nonstandard analysis"
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Thanks, Anome. But if we have separate "formal" proofs that:
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Wikipedia_talk:WikiProject_Mathematics#Article_product_rule
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Sometimes this idea is conveyed with a picture of areas.
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I'm not versed in non-standard analysis. What does the
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I couldn't understand this either and have removed it.
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There is another "product rule" that is often used in
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then does that make the proof count as a formal one?
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Isn't the law only valid for constant-mass systems?
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1881:
1867:
1841:
1800:
1762:
1754:
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1683:
1645:
1643:
1612:
1566:
1509:
1508:
1433:
1432:
1326:
1220:
1219:
1138:
1041:
1040:
948:
799:
797:
404:(Ο); likewise for
101:Mathematics portal
45:content assessment
3046:differential form
2908:comment added by
2891:Formatting error.
2853:The article says
2688:
2641:
2537:
2371:
2345:
1954:source for barrow
1866:
1840:
1799:
1753:
1728:
1682:
1642:
1611:
1565:
1324:
1295:
1251:
1136:
1110:
1072:
939:
245:
228:comment added by
166:
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3143:
3050:exterior product
2994:Joe in Australia
2920:
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2772:
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2752:
2751:
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2683:
2677:
2673:
2667:
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2649:
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2642:
2640:
2636:
2630:
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2620:
2554:
2552:
2551:
2546:
2538:
2536:
2532:
2526:
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2511:
2505:
2404:Tobias Bergemann
2382:
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1890:
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1798:
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1763:
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1730:
1727:
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1711:
1684:
1681:
1670:
1654:
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1651:
1646:
1644:
1641:
1633:
1625:
1613:
1610:
1602:
1594:
1567:
1564:
1553:
1518:
1516:
1515:
1510:
1500:
1483:
1472:
1442:
1440:
1439:
1434:
1430:
1407:
1396:
1339:make it easier.
1335:
1333:
1332:
1327:
1325:
1323:
1315:
1307:
1296:
1294:
1286:
1278:
1252:
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1239:
1229:
1227:
1226:
1221:
1217:
1194:
1183:
1147:
1145:
1144:
1139:
1137:
1135:
1127:
1119:
1111:
1109:
1101:
1093:
1073:
1071:
1060:
1050:
1048:
1047:
1042:
1038:
1021:
1010:
957:
955:
954:
949:
944:
940:
935:
900:
894:
856:
808:
806:
805:
800:
798:
791:
790:
760:
713:
675:
665:
661:
657:
656:
626:
595:
591:
587:
586:
556:
525:
473:
244:
222:
134:
133:
130:
127:
124:
103:
98:
97:
87:
80:
79:
74:
66:
59:
42:
33:
32:
25:
24:
16:
3151:
3150:
3146:
3145:
3144:
3142:
3141:
3140:
3091:
3090:
3070:
3026:
2989:
2948:
2926:
2903:
2899:
2893:
2851:
2795:
2794:
2757:
2756:
2734:
2733:
2678:
2668:
2661:
2660:
2631:
2621:
2608:
2607:
2527:
2506:
2493:
2492:
2472:
2398:st denotes the
2363:
2362:
2359:
2337:
2336:
2303:
2058:
1956:
1858:
1832:
1791:
1780:
1779:
1745:
1737:
1720:
1712:
1674:
1663:
1662:
1634:
1626:
1603:
1595:
1557:
1546:
1545:
1538:
1536:AssociativityΒ ?
1493:
1476:
1465:
1448:
1447:
1423:
1400:
1389:
1372:
1371:
1362:
1316:
1308:
1287:
1279:
1243:
1233:
1232:
1210:
1187:
1176:
1159:
1158:
1128:
1120:
1102:
1094:
1064:
1054:
1053:
1031:
1014:
1003:
989:
988:
982:
968:
901:
895:
849:
844:
843:
796:
795:
782:
753:
706:
670:
667:
666:
648:
619:
603:
599:
578:
549:
533:
529:
520:
517:
516:
468:
429:
428:
371:
339:
316:
302:Rule of product
251:
223:
220:
193:0, Ξ΄y/Ξ΄x -: -->
192:a) As Ξ΄x -: -->
171:
131:
128:
125:
122:
121:
99:
92:
72:
43:on Knowledge's
40:
30:
12:
11:
5:
3149:
3147:
3139:
3138:
3133:
3128:
3123:
3118:
3113:
3108:
3103:
3093:
3092:
3069:
3066:
3065:
3064:
3025:
3022:
3021:
3020:
2988:
2985:
2947:
2942:
2925:
2922:
2897:
2892:
2889:
2888:
2887:
2850:
2847:
2846:
2845:
2844:
2843:
2842:
2841:
2840:
2839:
2838:
2837:
2814:
2811:
2808:
2805:
2802:
2778:
2775:
2771:
2767:
2764:
2744:
2741:
2721:
2720:
2719:
2718:
2717:
2716:
2715:
2714:
2702:
2701:
2700:
2686:
2682:
2676:
2672:
2655:
2654:
2653:
2639:
2635:
2629:
2625:
2618:
2615:
2595:
2594:
2593:
2592:
2591:
2590:
2570:
2569:
2557:
2556:
2555:
2544:
2541:
2535:
2531:
2525:
2521:
2517:
2514:
2510:
2503:
2500:
2471:
2468:
2467:
2466:
2465:
2464:
2463:
2462:
2446:. The reason
2435:
2434:
2433:
2432:
2415:
2414:
2358:
2334:
2333:
2332:
2302:
2299:
2298:
2297:
2296:
2295:
2294:
2293:
2292:
2291:
2290:
2289:
2288:
2287:
2286:
2285:
2284:
2283:
2227:
2226:
2225:
2224:
2223:
2222:
2221:
2220:
2219:
2218:
2194:
2193:
2192:
2191:
2190:
2189:
2188:
2187:
2165:
2164:
2163:
2162:
2161:
2160:
2159:
2158:
2137:
2136:
2135:
2134:
2133:
2132:
2118:
2109:
2108:
2107:
2106:
2089:
2088:
2057:
2054:
2053:
2052:
2051:
2050:
2049:
2048:
2047:
2046:
2024:
2023:
2022:
2021:
2020:
2019:
1987:
1986:
1955:
1952:
1951:
1950:
1931:
1930:
1892:
1891:
1880:
1877:
1874:
1871:
1864:
1861:
1857:
1851:
1848:
1845:
1838:
1835:
1831:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1797:
1794:
1790:
1773:
1772:
1761:
1758:
1751:
1748:
1743:
1740:
1733:
1726:
1723:
1718:
1715:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1680:
1677:
1673:
1656:
1655:
1640:
1637:
1632:
1629:
1622:
1619:
1616:
1609:
1606:
1601:
1598:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1570:
1563:
1560:
1556:
1537:
1534:
1520:
1519:
1506:
1503:
1499:
1496:
1492:
1489:
1486:
1482:
1479:
1475:
1471:
1468:
1464:
1461:
1458:
1455:
1429:
1426:
1422:
1419:
1416:
1413:
1410:
1406:
1403:
1399:
1395:
1392:
1388:
1385:
1382:
1379:
1368:... that isΒ :
1361:
1358:
1356:
1337:
1336:
1322:
1319:
1314:
1311:
1305:
1302:
1299:
1293:
1290:
1285:
1282:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1249:
1246:
1242:
1230:
1216:
1213:
1209:
1206:
1203:
1200:
1197:
1193:
1190:
1186:
1182:
1179:
1175:
1172:
1169:
1166:
1149:
1148:
1134:
1131:
1126:
1123:
1117:
1114:
1108:
1105:
1100:
1097:
1091:
1088:
1085:
1082:
1079:
1076:
1070:
1067:
1063:
1051:
1037:
1034:
1030:
1027:
1024:
1020:
1017:
1013:
1009:
1006:
1002:
999:
996:
984:Formulas like
981:
978:
966:
959:
958:
947:
943:
938:
934:
931:
928:
925:
922:
919:
916:
913:
910:
907:
904:
898:
893:
890:
887:
884:
881:
878:
875:
872:
868:
865:
862:
859:
855:
852:
810:
809:
794:
789:
785:
781:
778:
775:
772:
769:
766:
763:
759:
756:
752:
749:
746:
743:
740:
737:
734:
731:
728:
725:
722:
719:
716:
712:
709:
705:
702:
699:
696:
693:
690:
687:
684:
681:
678:
673:
671:
669:
668:
664:
660:
655:
651:
647:
644:
641:
638:
635:
632:
629:
625:
622:
618:
615:
612:
609:
606:
602:
598:
594:
590:
585:
581:
577:
574:
571:
568:
565:
562:
559:
555:
552:
548:
545:
542:
539:
536:
532:
528:
523:
521:
519:
518:
515:
512:
509:
506:
503:
500:
497:
494:
491:
488:
485:
482:
479:
476:
471:
469:
467:
464:
461:
458:
455:
452:
449:
446:
443:
440:
437:
436:
370:
367:
366:
365:
355:
354:
338:
335:
328:
327:
323:
315:
312:
250:
249:Disambiguation
247:
218:
213:
205:
195:
170:
167:
164:
163:
160:
159:
156:
155:
144:
138:
137:
135:
118:the discussion
105:
104:
88:
76:
75:
67:
55:
54:
48:
26:
13:
10:
9:
6:
4:
3:
2:
3148:
3137:
3134:
3132:
3129:
3127:
3124:
3122:
3119:
3117:
3114:
3112:
3109:
3107:
3104:
3102:
3099:
3098:
3096:
3089:
3088:
3084:
3080:
3074:
3067:
3063:
3059:
3055:
3051:
3047:
3043:
3042:
3041:
3040:
3036:
3032:
3019:
3015:
3011:
3006:
3005:
3004:
3003:
2999:
2995:
2986:
2984:
2983:
2979:
2975:
2972:
2968:
2967:
2965:
2961:
2957:
2953:
2946:
2943:
2941:
2940:
2936:
2932:
2923:
2921:
2919:
2915:
2911:
2910:110.20.69.222
2907:
2896:
2890:
2886:
2882:
2878:
2874:
2873:
2872:
2871:
2867:
2863:
2859:
2856:
2848:
2836:
2832:
2828:
2812:
2806:
2803:
2800:
2792:
2776:
2769:
2765:
2742:
2731:
2730:
2729:
2728:
2727:
2726:
2725:
2724:
2723:
2722:
2713:
2710:
2707:
2703:
2684:
2674:
2659:
2658:
2656:
2637:
2627:
2616:
2613:
2606:
2605:
2603:
2602:
2601:
2600:
2599:
2598:
2597:
2596:
2589:
2585:
2581:
2576:
2575:
2574:
2573:
2572:
2571:
2568:
2565:
2562:
2558:
2542:
2539:
2533:
2523:
2515:
2512:
2501:
2498:
2491:
2490:
2488:
2487:
2486:
2485:
2481:
2477:
2469:
2461:
2457:
2453:
2449:
2445:
2441:
2440:
2439:
2438:
2437:
2436:
2431:
2427:
2423:
2419:
2418:
2417:
2416:
2413:
2409:
2405:
2401:
2397:
2396:
2395:
2394:
2390:
2386:
2331:
2327:
2323:
2319:
2318:
2317:
2316:
2312:
2308:
2300:
2282:
2278:
2274:
2269:
2268:
2267:
2263:
2259:
2254:
2253:
2252:
2248:
2244:
2239:
2238:
2237:
2236:
2235:
2234:
2233:
2232:
2231:
2230:
2229:
2228:
2217:
2213:
2209:
2204:
2203:
2202:
2201:
2200:
2199:
2198:
2197:
2196:
2195:
2186:
2182:
2178:
2173:
2172:
2171:
2170:
2169:
2168:
2167:
2166:
2157:
2153:
2149:
2145:
2144:
2143:
2142:
2141:
2140:
2139:
2138:
2131:
2127:
2123:
2119:
2115:
2114:
2113:
2112:
2111:
2110:
2105:
2101:
2097:
2093:
2092:
2091:
2090:
2087:
2083:
2079:
2075:
2074:
2073:
2072:
2068:
2064:
2055:
2045:
2041:
2037:
2032:
2031:
2030:
2029:
2028:
2027:
2026:
2025:
2018:
2014:
2010:
2005:
2004:
2003:
1999:
1995:
1991:
1990:
1989:
1988:
1985:
1981:
1977:
1972:
1971:
1970:
1969:
1965:
1961:
1953:
1949:
1945:
1941:
1937:
1933:
1932:
1929:
1925:
1921:
1920:Michael Hardy
1917:
1913:
1909:
1908:
1907:
1906:
1902:
1898:
1878:
1875:
1869:
1862:
1859:
1855:
1846:
1843:
1836:
1833:
1829:
1823:
1820:
1817:
1811:
1808:
1805:
1795:
1792:
1788:
1778:
1777:
1776:
1759:
1756:
1749:
1746:
1741:
1738:
1731:
1724:
1721:
1716:
1713:
1706:
1703:
1700:
1694:
1691:
1688:
1678:
1675:
1671:
1661:
1660:
1659:
1638:
1635:
1630:
1627:
1620:
1617:
1614:
1607:
1604:
1599:
1596:
1589:
1586:
1583:
1577:
1574:
1571:
1561:
1558:
1554:
1544:
1543:
1542:
1535:
1533:
1532:
1528:
1524:
1504:
1501:
1497:
1494:
1490:
1487:
1484:
1480:
1477:
1473:
1469:
1462:
1459:
1456:
1446:
1445:
1444:
1427:
1424:
1420:
1417:
1414:
1411:
1408:
1404:
1401:
1397:
1393:
1386:
1383:
1380:
1369:
1366:
1359:
1357:
1354:
1352:
1348:
1344:
1340:
1320:
1317:
1312:
1309:
1303:
1300:
1297:
1291:
1288:
1283:
1280:
1274:
1271:
1268:
1262:
1259:
1256:
1247:
1244:
1240:
1231:
1214:
1211:
1207:
1204:
1201:
1198:
1195:
1191:
1188:
1184:
1180:
1173:
1170:
1167:
1157:
1156:
1155:
1152:
1132:
1129:
1124:
1121:
1115:
1112:
1106:
1103:
1098:
1095:
1089:
1086:
1080:
1077:
1068:
1065:
1061:
1052:
1035:
1032:
1028:
1025:
1022:
1018:
1015:
1011:
1007:
1000:
997:
987:
986:
985:
979:
977:
976:
973:
965:
962:
945:
941:
936:
929:
923:
920:
914:
911:
908:
902:
896:
866:
860:
853:
850:
842:
841:
840:
838:
833:
831:
827:
823:
819:
815:
787:
783:
776:
773:
770:
764:
757:
754:
747:
741:
738:
735:
729:
723:
717:
710:
707:
703:
697:
691:
685:
679:
676:
672:
662:
653:
649:
642:
639:
636:
630:
623:
620:
616:
610:
604:
600:
596:
592:
583:
579:
572:
569:
566:
560:
553:
550:
546:
540:
534:
530:
526:
522:
510:
507:
504:
498:
495:
489:
486:
483:
477:
474:
470:
462:
459:
456:
447:
444:
441:
427:
426:
425:
423:
419:
415:
411:
407:
403:
399:
395:
391:
387:
383:
379:
374:
368:
364:
361:
357:
356:
353:
350:
346:
345:
344:
343:
336:
334:
332:
324:
321:
320:
319:
313:
311:
310:
307:
303:
299:
298:combinatorics
294:
292:
288:
282:
281:
276:
274:
270:
266:
261:
260:
256:
248:
246:
243:
239:
235:
231:
227:
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18:
17:
3079:Kontribuanto
3075:
3071:
3027:
2990:
2969:
2949:
2927:
2904:β Preceding
2900:
2894:
2854:
2852:
2790:
2706:Arthur Rubin
2561:Arthur Rubin
2473:
2422:67.158.43.41
2385:67.158.43.41
2360:
2304:
2059:
1957:
1935:
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1911:
1897:213.68.42.99
1893:
1774:
1657:
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1521:
1370:
1367:
1363:
1355:
1341:
1338:
1153:
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969:
963:
960:
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421:
417:
413:
409:
405:
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397:
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389:
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375:
372:
341:
340:
329:
317:
295:
291:Tim Starling
286:
283:
280:Pizza Puzzle
277:
269:product rule
262:
255:Product rule
252:
224:β Preceding
214:
206:
201:Kidburla2002
198:
191:
188:
185:
175:
172:
148:Mid-priority
147:
107:
73:Midβpriority
51:WikiProjects
34:
412:+Ο). Then (
123:Mathematics
114:mathematics
70:Mathematics
3095:Categories
2931:Michealefr
326:necessary?
1658:but only
1443:and not
1343:Bo Jacoby
273:AxelBoldt
210:The Anome
179:PAStheLoD
39:is rated
3054:D.Lazard
3031:Rdsk2014
2906:unsigned
2357:notation
349:Revolver
331:Revolver
306:Hughitt1
238:contribs
226:unsigned
2791:defines
2301:Example
384:+Ο) as
360:Eric119
300:. (see
259:Tarquin
230:Natkuhn
150:on the
41:B-class
3010:Mgnbar
2974:Dennui
2956:Dennui
2877:Adpete
2862:Adpete
2827:Tkuvho
2789:, one
2709:(talk)
2580:Tkuvho
2564:(talk)
2476:Tkuvho
2452:Tkuvho
2322:Tkuvho
2273:Tkuvho
2243:Tkuvho
2177:Tkuvho
2148:Tkuvho
2096:Tkuvho
2063:Tkuvho
2036:Tkuvho
1994:Tkuvho
1960:Tkuvho
47:scale.
1523:7yl4r
972:KSmrq
257:? --
169:Proof
28:This
3083:talk
3058:talk
3035:talk
3014:talk
2998:talk
2978:talk
2960:talk
2935:talk
2914:talk
2881:talk
2866:talk
2831:talk
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2480:talk
2456:talk
2426:talk
2408:talk
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2326:talk
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2277:talk
2262:talk
2258:Dmcq
2247:talk
2212:talk
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2181:talk
2152:talk
2126:talk
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2100:talk
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2078:Dmcq
2067:talk
2040:talk
2013:talk
2009:Dmcq
1998:talk
1980:talk
1976:Dmcq
1964:talk
1944:talk
1940:Dmcq
1924:talk
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1347:talk
234:talk
186:---
173:Hi,
2825:.
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337:???
287:the
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1275:β
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1196:β
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937:Ο
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771:Ο
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650:Ο
637:Ο
597:β
580:Ο
567:Ο
511:Ο
496:β
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396:β²(
392:)+
271:.
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53::
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