95:
1494:
85:
64:
31:
1258:
4361:). Having repetitions in both notations makes the page less compact and reading more difficult (this is, after all, not a tutorial on the equivalence of the notations); how to reorganize this sensibly is not obvious to me. I imagine that it is common even at undergraduate level to switch freely between these notations according to convenience (compactness vs. algebraic suggestiveness), complicating ths choice.
5778:
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22:
3348:
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1489:{\displaystyle {\frac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}={\frac {\mathrm {d} }{\mathrm {d} x}}\left(\cdots {\frac {\mathrm {d} }{\mathrm {d} x}}\left({\frac {\mathrm {d} y}{\mathrm {d} x}}\right)\right)={\frac {\mathrm {d} \cdots \mathrm {d} \mathrm {d} y}{\mathrm {d} x\cdots \mathrm {d} x\mathrm {d} x}}={\frac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}}
3161:
4548:(or higher derivatives) appears on its own is problematic, as this notation for higher derivatives does not make algebraic sense (according to Roger Penrose), even though it does at the level of first derivatives. Given this, it may make sense to stick with the notation where it is presented as ratios or operators throughout, e.g.
5482:
4740:(lower) for consistency with the previous setion. (Honestly - i've always seen and used capital and lower case for functions interchangably - capitalized letters of a function don't always mean a Fourier transform of the lower case function, though it is a convention, sometimes for a FT tildes or hats are used).
3172:
4428:
Furthermore, why is it such a big problem to define nomeclature then proceed through the rest of the article? I've done this a number of times and everyone thinks its unclear. I don't mind that, but why do people thinks it unclear and useless?. How are we supposed to know what is what? Are we really
1247:
2174:
1866:
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4931:
But you also removed the rules for differentiation of composite and indirect functions which I think should be restored here (and I think we also could add formulas for differentiation of multivariable functions, differentiation of a function in certain direction
2927:
4921:
I think arithmetic calculations made by CAS or a calculator are not an original research, making such calculations is permitted by the rules. But this is anyway related to the formulas for higher-order derivatives for which we can have another list (for example,
3010:
4103:
I object, I just didn't notice the comment made a week ago. The current article uses both notations, and it should. Words are generally preferred to formulas for the purposes of learning, but I do not object stenuously to using function notation on occasion.
5773:{\displaystyle {\begin{aligned}{\frac {d}{dx}}\left(\ln |x|\right)\\={\frac {d}{dx}}\left(\ln u\right)\\={\frac {1}{u}}{\frac {du}{dx}}\\={\frac {1}{|x|}}{\frac {d}{dx}}\left(|x|\right)\\={\frac {1}{|x|}}{\frac {x}{|x|}}\\={\frac {x}{{|x|}^{2}}}\end{aligned}}}
3455:
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5848:
Some pages link to almost no other languages at all and the content even if should be the same is totally different most likely cause some authors can't observe the article in other languages in order to inspire from much better written articles.
5932:
3343:{\displaystyle \int {\frac {\mathrm {d} (pq)}{\mathrm {d} t}}\mathrm {d} t=\int p\mathrm {d} q+\int q\mathrm {d} p=\int {\frac {p\mathrm {d} q}{\mathrm {d} t}}\mathrm {d} t+\int {\frac {q\mathrm {d} p}{\mathrm {d} t}}\mathrm {d} t\,\!}
4274:
Any expansion to wider areas of applicability such as complex numbers, vectors and matrices (potentially very extensive) should be kept off this page. Any such expansions should be in other similar pages, referenced in a "see also"
1851:
1716:
3782:
prime notation in this article and use the almighty tools of
Leibnitz: the d, and the ∫. Anyone reading this will think I am being sooo pedantic and conscious about notation. But consciousness of powerful notation is a good thing.
2466:{\displaystyle {\frac {\mathrm {d} y}{\mathrm {d} x}}={\frac {\lim _{\delta y\to 0^{-}}\delta y}{\lim _{\delta x\to 0}\delta x}}={\frac {\lim _{\delta x\to 0^{-}}}{\lim _{\delta x\to 0}\delta x}}=\lim _{\delta x\to 0^{-}}\left\,}
2158:{\displaystyle {\frac {\mathrm {d} y}{\mathrm {d} x}}={\frac {\lim _{\delta y\to 0^{+}}\delta y}{\lim _{\delta x\to 0}\delta x}}={\frac {\lim _{\delta x\to 0^{+}}}{\lim _{\delta x\to 0}\delta x}}=\lim _{\delta x\to 0^{+}}\left\,}
3657:
858:
it is unclear what to differentiate with respect to, context needs to provide explanation instead of stating the variable. With d notation only the variables and their symbols need to be stated - then its absolutely clear what
4885:. I feel that this content is particularly unsuited to this article, since they aren't considered to be "differentiation rules" in the same sense that the other rules listed here are. It has been suggested that the original
2635:
5471:
1116:
1008:
4904:). Now, I have no doubt that a reference could be found for some of these, but so can references for millions of trivial differentiation identities. We are not a repository for that kind of indiscriminate information.
4270:
I think the page should remain as focused on essentially the rules one learns first: relating to real functions of one real variable. If need be, the article could be renamed to reflect this focus on single-variable real
2567:
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5001:
PS - good job to everyone who re-wrote the article, since I effectivley screwed it up. Its really clear from months ago. I don't favour or care about the annihilation of my own creation of the N diffs table, of
4046:
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4954:
as this is commonly understood in textbooks on the subject. I agree that perhaps a mention of the chain rule in several variables might be a good idea, but I wouldn't emphasize too many special cases of it.
3002:
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4853:
There has been a proposal to restore this section, it may as well be launched back into the article. Any article benefits from relavant sources/refs. Also I added a few relevant extra articles under the
3876:
2933:
It makes integration much easier, to see which infinitesimal quantities cancel when integrating a derivative. It makes integration by parts and by substitution trillions of times more obvious and easier.
4243:'s earlier comments: I didn't have this article on my watchlist, and I suspect a lot of other people didn't either. If you're making a major overhaul to an article, it's a good idea to mention it at
2641:
4647:– this is in keeping with Leibnitz's notation being suggestive only. We should in any event not be going out on a limb with the notation if standard texts don't do this. You may be interested in
5487:
3156:{\displaystyle {\frac {\mathrm {d} (pq)}{\mathrm {d} t}}={\frac {p\mathrm {d} q+q\mathrm {d} p}{\mathrm {d} t}}={\frac {p\mathrm {d} q}{\mathrm {d} t}}+{\frac {q\mathrm {d} p}{\mathrm {d} t}}\,}
846:
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1516:. The manipulation of multiplying/dividing by infinitesimal quantities can be just that and allow all derivatives to be derived. To derive the ratio of infinitesimal amounts of variables
5991:
3775:
Just look at the notation alive in action - it’s on the loose, dynamic, manipulating, resembles the familiar elementary algebra, and implies a direct formalism with linear algebra...
3356:
4707:
and the already italic d in this article, I suppose the nth diff section should have d italicised. (b.t.w. I wrote that table - others edited after without actually deleting it). --
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A definitions/notations section of more than a sentence or two is a liability. The change so that the first view of the page as full of abstract symbols definitely is a problem.
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section has been completley emptied... I'll try to find sources for the core rules to fill in at least a couple of cited referances, in addition than those in the
4267:
The article seems to be intended as a "quick reference" (like an appendix of Shaum's
Outline Series), not as a rigorous guide of applicable domains and the like.
4455:
clarification introduced. And I didn't not oppose your original point about notation. You unfortunately included unrelated contentious edits at the same time.
35:
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is, the infinitesimal amount simply has a d in front, using subscripts just clutters notation, which should be used for index variables or naming variables,
5986:
1242:{\displaystyle (\mathrm {d} x)(\mathrm {d} x)\cdots (\mathrm {d} x)=\mathrm {d} x\mathrm {d} x\cdots \mathrm {d} x=(\mathrm {d} x)^{n}=\mathrm {d} x^{n}\,}
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4421:. I had other things to do (and no it wasn't sleep as you'll assume - writing up solutions to assessements for uni was one of them...). Again - sorry ×
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I truley apologize for not staying up all night staring my eyeballs out at the screen waiting for the off-chance that people will reply, then
4131:
The recent series of edits has many problems. In the first section, "continuous where specified" is irrelevant, as is emphasizing a notation
5193:
5094:
405:{\displaystyle {d^{n} \over dx^{n}}f(g(x))=\sum _{\pi \in \Pi }f^{(\left|\pi \right|)}(g(x))\cdot \prod _{B\in \pi }g^{(\left|B\right|)}(x)}
5856:
4239:. An encyclopedia article should conform to the commonly used notation, even if it is less than ideal. Regarding the lack of response to
4544:
I'll be presumptious and answer that from my perspective (Sławomir Biały may have meant something different). To use a notation in which d
885:
4193:: this is very problematic in this context. It is tempting to go back to the last sane version. What, exactly, is worth keeping here?
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must be equal for continuity. Prime notation hides this: the ratio of two infinitesimal quantities is more general than a derivative. d
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its problematic, the sentence simply ended there...why? (here's you're chance to look good and make me seem more stupid than ever)--
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The domain of applicability can be left to be inferred as it so often is; no more than a simple statement to this effect is needed.
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2736:{\displaystyle \mathrm {d} \ell ={\sqrt {\mathrm {d} x^{2}+{\frac {\mathrm {d} x^{2}}{\mathrm {d} x^{2}}}\mathrm {d} y^{2}}}\,}
759:
878:, but is a correct interpretation: d can be considered as a very very very (...) small number tending asymptotically to zero:
5290:
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4281:
The change of function names to uppercase also is unusual; it creates the impression that something specific is meant (e.g.
4222:
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5803:
isn't sinh(x) = (e^x - e^-x)/2, and if so, then the derivative of cosh(x) should equal positive sinh(x) not the negative.
4900:
to me. Not only that, but it seems to be sourced entirely to someone's empirical observations from a CAS (which makes it
4750:
this sentence is a bit funny - I don't remember writing it this way all that time ago (though probably my mistake anyway):
4661:
4461:
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44:
4432:
It doesn't matter - enjoy your prime notation... No revert to d will ever be made. No I couldn't care less if all you
2922:{\displaystyle L=\int \mathrm {d} \ell =\int {\sqrt {1+{\frac {\mathrm {d} y^{2}}{\mathrm {d} x^{2}}}}}\mathrm {d} x\,}
5036:
4364:
It may make sense to limit edits to one type of change at a time; that way other editors can follow what is happening.
4923:
4404:'s earlier comments: I didn't have this article on my watchlist, and I suspect a lot of other people didn't either."
1500:
the notion of an infinitesimal quantity is too powerful to ignore. The ratio of two infinitesimal quantities (say) d
781:
226:
3450:{\displaystyle \int \mathrm {d} (pq)=\int p\mathrm {d} q+q\mathrm {d} p=\int p\mathrm {d} q+\int q\mathrm {d} p\,}
5056:
I suppose it's defined for x < 0 for complex numbers? Since there's no notation of x < 0 not allowed in R.
4956:
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5822:
The languages menu for the differentiation is really messed up because in some languages it's called derivate
4897:
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It might be best to decompose into bullet points instead of evrything all on one line. This much can be done.--
4055:
I suppose someone will still merrily delete my edits after, even with this much justification for an effort to
5860:
21:
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1101:{\displaystyle \mathrm {d} (\cdots (\mathrm {d} (\mathrm {d} x)))=\mathrm {d\cdots dd} x=\mathrm {d} ^{n}x\,}
453:
5899:
5073:
442:
shouldn't the n'th derivative of the power x^N be proportional to x^(N-n) instead of the product index r ?
427:
3765:{\displaystyle \int y(x)\mathrm {d} x=\int y(x(t)){\frac {\mathrm {d} x(t)}{\mathrm {d} t}}\mathrm {d} t\,}
2823:{\displaystyle \mathrm {d} \ell =\mathrm {d} x{\sqrt {1+{\frac {\mathrm {d} y^{2}}{\mathrm {d} x^{2}}}}}\,}
5088:
I believe that there's another error in the derivative of ln |x|. The current version of this page says:
5032:
4793:
4655:
is not an infinitesimal, and behaves algebraically as for
Leibnitz (but you can't divide by it, and d=0).
4950:
The idea is not to have a list of every conceivable derivative that could arise, but rather to cover the
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on
February 6, 2011. For the contribution history and old versions of the redirected page, please see
5852:
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Why doesn't the article mention that trig functions must be in radians not degrees for this to work!
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This belongs in the Nth derivatives section, but someone who knows more about math needs to do this.
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as the inverse when used. This fits with the use of the isolated function name as a variable (e.g.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
5923:
This article was the subject of a Wiki
Education Foundation-supported course assignment, between
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3829:" .....) by collecting base formulae (product/quotient/chain/general power rulse etc) together,
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In the end I kept both notations. Even still, will you delete the previous two edits I made?--
4066:
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to clarify what the function must be for the derivative to exist. E.x. cannot have a function
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to consider. And you come off as a little crazy in your post, if you don't mind my saying.
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i'm going cut down the monotonous repetetive writing in each subsection (e.x. "for functions
1110:
We can also multiply the full dx with itself as many times as needed, raising it to a power:
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If you're responding to my points, perhaps you didn't read too well. I was saying there was
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its easy to loose this when writing for a dot, splodge, smudge etc. and may lead to errors,
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must be equal for continuity, the right-hand ratio of two quantities (also a derivative)
5948:
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are, if they can quickly be tucked under the d, integration by parts is really quick.
1846:{\displaystyle \mathrm {d} y=\lim _{\delta y\to 0}\delta y=\lim _{\delta x\to 0^{-}}\,}
1711:{\displaystyle \mathrm {d} y=\lim _{\delta y\to 0}\delta y=\lim _{\delta x\to 0^{+}}\,}
525:
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4334:)); that way it is unnecessary to clarify the notation, except to note the exception
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However, I'm not an expert in maths so was wondering if someone could verify this:
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3652:{\displaystyle \mathrm {d} x={\frac {\mathrm {d} x}{\mathrm {d} t}}\mathrm {d} t\,}
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2630:{\displaystyle \mathrm {d} \ell ={\sqrt {\mathrm {d} x^{2}+\mathrm {d} y^{2}}}\,}
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5466:{\displaystyle {\frac {d}{dx}}\left(\ln u\right)={\frac {1}{u}}{\frac {du}{dx}}}
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4061:, not complicate. Please at least see where i'm coming from before doing so.--
1003:{\displaystyle \mathrm {d} (\mathrm {d} x)=\mathrm {dd} x=\mathrm {d} ^{2}x\,}
222:
90:
1012:"Multiplying" by d any number of times gives an even smaller still quantity:
2562:{\displaystyle \mathrm {d} \ell ^{2}=\mathrm {d} x^{2}+\mathrm {d} y^{2}\,}
5838:
4881:. It was recently brought to my attention that this was merged here from
4809:
775:
This is not just my personal comment. Prime notation for derivatives is:
5843:
5276:{\displaystyle {\frac {d}{dx}}\left(\ln |x|\right)={\frac {x}{|x|^{2}}}}
5177:{\displaystyle {\frac {d}{dx}}\left(\ln |x|\right)={\frac {|x|}{x^{2}}}}
5084:
Antoher Error in 'Derivatives of exponential and logarithmic functions'?
4476:
of you - some of the dominant editors of maths articles (hence the term
2496:(in 2d Cartesian plane) can be written directly as Pythagoras' theorem:
3832:
write the definitions of the function in the standard formalism (e.x.):
4429:
supposed to keep explaining symbols throughout? I know - its just me.
930:{\displaystyle \mathrm {d} x=\lim _{\delta x\rightarrow 0}\delta x\,}
5083:
5045:
3977:{\displaystyle f:\mathbf {R} -\{x_{0}\}\rightarrow \mathbf {R} \,\!}
2492:
x can be treated as algebraic quantities. Ex. the length of a curve
4679:
Very well. Now is probably the best time to close this section. --
5875:
Why there is no inverse rules main important rules are onlt they
4041:{\displaystyle f\circ g:\mathbf {R} \rightarrow \mathbf {R} \,\!}
5369:{\displaystyle {\frac {d}{dx}}\left(|x|\right)={\frac {x}{|x|}}}
5046:
error in 'Derivatives of exponential and logarithmic functions'?
2997:{\displaystyle \mathrm {d} (pq)=p\mathrm {d} q+q\mathrm {d} p\,}
4896:
I personally don't think it should be, because it seems pretty
5833:
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section, not already linked into the context of the article.--
167:
15:
5825:
Please take a close look at the language menu of these pages
1576:{\displaystyle \mathrm {d} x=\lim _{\delta x\to 0}\delta x\,}
5828:
4877:
I recently removed some of the content of this article with
4289:). The notation thus being less familiar is not a good idea.
3800:
Wow - no one has objected after a week. I'll go for it. Also
3512:{\displaystyle pq=\int p\mathrm {d} q+\int q\mathrm {d} p\,}
2168:
left-hand ratio of two quantities (again also a derivative)
940:"Multiplying" by another d gives an even smaller quantity: :
4484:
think the d notation is so "problematic" in this context?
3871:{\displaystyle f:\mathbf {R} \rightarrow \mathbf {R} \,\!}
4893:
so that this content has a place somewhere. Opinions?
4472:
Now I apologize for not being clearer. I was talking to
4235:
should be restored. But in all other respects I prefer
4052:(I h8 this circle notation - it also looks stupid.....).
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Putting these two together, we set u to be |x|, then:
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However, I believe that the correct answer should be:
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Derivatives of exponential and logarithmic functions,
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There may be a case for limiting mixing of notation (d
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https://fr.wikipedia.org/D%C3%A9riv%C3%A9es_usuelles
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4315:)) (never the operator-like notation for functions
4231:I think the "sources and further reading" added in
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Eliminate prime notation - use
Leibnitz' powerful d
5914:Wiki Education assignment: 4A Knowledge Assignment
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4436:think fucktards like me are crazy. Irrelavent.--
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5892:Differentiation rules#The inverse function rule
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5839:https://en.wikipedia.org/Differentiation_rules
4162:is irrelevant. Get rid of "Leibniz notation"
841:{\displaystyle f'=\mathrm {effing\,stupid} \,}
8:
4520:: this is very problematic in this context."
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874:more intuitive: this is not actually what d
198:; for the discussion at that location, see
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5834:https://ro.wikipedia.org/Tabel_de_derivate
215:List of List of differentiation identities
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209:completed merge, some assistance required
5829:https://es.wikipedia.org/Anexo:Derivadas
2937:The differential product rule is clear:
463:Thanks very much for finding that ! ...
5982:Knowledge vital articles in Mathematics
4034:
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478:The polynomial or elementary power rule
60:
5799:possible error in hyperbolic functions
4245:Knowledge talk:WikiProject Mathematics
4926:) or include it in a related article.
3166:hence integration by parts is clear:
7:
1508:is either that or the derivative of
106:This article is within the scope of
5062:but no notion of x =/= 0 is given
4480:). Also I forgot to ask - why does
4400:"Regarding the lack of response to
482:Right now this section states: "If
49:It is of interest to the following
5987:Start-Class level-5 vital articles
5928:
5924:
4887:list of differentiation identities
4883:list of differentiation identities
4522:<---There was no mention as to
4211:I have asked for further input at
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229:(below is the combinatoric form):
181:List of differentiation identities
14:
6007:Mid-priority mathematics articles
648:{\displaystyle n\in \mathbb {C} }
620:{\displaystyle n\in \mathbb {N} }
588:{\displaystyle f'(x)=nx^{n-1}.\,}
126:Knowledge:WikiProject Mathematics
6002:Start-Class mathematics articles
5977:Knowledge level-5 vital articles
5931:. Further details are available
5918:
4513:{\displaystyle df={\text{blah}}}
4489:""Get rid of "Leibniz notation"
4186:{\displaystyle df={\text{blah}}}
4030:
4022:
3966:
3939:
3860:
3852:
171:
129:Template:WikiProject Mathematics
93:
83:
62:
29:
20:
5890:Do you mean this inverse rule:
4924:List of higher-oder derivatives
4285:being the Fourier transform of
3587:{\displaystyle y=y(x),x=x(t)\,}
3533:Integration by sub is obvious:
146:This article has been rated as
5813:13:52, 30 September 2015 (UTC)
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4849:Sources and further reading...
4757:"...real number constants are
4743:The title can be made better:
4350:)) rather than as an operator.
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5947:— Assignment last updated by
5865:22:30, 22 November 2015 (UTC)
5041:00:14, 16 February 2012 (UTC)
4916:11:48, 30 December 2011 (UTC)
3795:18:51, 21 November 2011 (UTC)
599:But this isn't only true for
120:and see a list of open tasks.
5908:18:08, 9 February 2018 (UTC)
5885:17:57, 9 February 2018 (UTC)
5794:00:43, 19 October 2013 (UTC)
4868:09:58, 3 December 2011 (UTC)
4844:09:19, 3 December 2011 (UTC)
4732:(capital) need to change to
4717:08:37, 3 December 2011 (UTC)
4689:08:37, 3 December 2011 (UTC)
4667:20:43, 2 December 2011 (UTC)
4536:18:49, 2 December 2011 (UTC)
4467:11:24, 2 December 2011 (UTC)
4446:08:22, 2 December 2011 (UTC)
4380:06:23, 2 December 2011 (UTC)
4257:05:32, 2 December 2011 (UTC)
4227:02:52, 2 December 2011 (UTC)
4205:02:48, 2 December 2011 (UTC)
4126:00:25, 2 December 2011 (UTC)
4095:03:17, 2 December 2011 (UTC)
4071:17:42, 1 December 2011 (UTC)
3923:), so the mapping would be:
3778:As such I'm going to remove
1252:this explains the notation:
736:{\displaystyle \mathbb {Z} }
714:{\displaystyle \mathbb {Q} }
692:{\displaystyle \mathbb {N} }
670:{\displaystyle \mathbb {R} }
433:17:39, 6 February 2011 (UTC)
5012:12:54, 6 January 2012 (UTC)
4997:12:43, 6 January 2012 (UTC)
4967:23:00, 5 January 2012 (UTC)
4942:20:16, 5 January 2012 (UTC)
4295:Keep the power notation to
764:20:54, 16 August 2011 (UTC)
221:For higher derivatives the
6023:
5972:Start-Class vital articles
5957:17:57, 22 April 2024 (UTC)
5379:|x|´ = x / |x| for x != 0
517:{\displaystyle f(x)=x^{n}}
458:11:20, 11 April 2011 (UTC)
5053:(ln x)´ = 1/x , x =/= 0.
4952:rules for differentiation
770:
145:
78:
57:
5382:Meanwhile, according to
5078:18:49, 28 May 2012 (UTC)
4812:(used for indicies) are
4745:Derivatives to nth order
3912:calculate the diff of 1/
473:12:20, 8 June 2011 (UTC)
152:project's priority scale
531:(including zero) then
109:WikiProject Mathematics
5774:
5467:
5370:
5277:
5178:
5029:Hardly a minor point!
4514:
4419:immediatley responding
4187:
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2476:again left and right d
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227:Faà di Bruno's formula
217:here except for this:
5935:. Student editor(s):
5775:
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4151:{\displaystyle F^{n}}
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190:Differentiation rules
178:The contents of the
36:level-5 vital article
5483:
5393:
5300:
5194:
5095:
4773:, real integers are
4493:
4166:
4135:
4006:
3988:for for composition
3929:
3890:) which vanishes at
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627:; it's true for any
603:
538:
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236:
132:mathematics articles
5022:Radians and degrees
4703:In accordance with
5933:on the course page
5770:
5768:
5463:
5366:
5273:
5174:
4873:Removal of content
4649:differential forms
4510:
4183:
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3821:", "for functions
3813:", "for functions
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101:Mathematics portal
45:content assessment
5871:About the inverse
5867:
5855:comment added by
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5033:Rosa Lichtenstein
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5925:12 February 2024
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5767:
5766:
5742:
5731:
5730:
5713:
5691:
5680:
5679:
5660:
5656:
5646:
5624:
5613:
5612:
5602:
5594:
5576:
5575:
5560:
5556:
5546:
5535:
5534:
5509:
5505:
5495:
5481:
5480:
5453:
5445:
5415:
5411:
5401:
5391:
5390:
5349:
5322:
5318:
5308:
5298:
5297:
5258:
5249:
5216:
5212:
5202:
5192:
5191:
5162:
5147:
5117:
5113:
5103:
5093:
5092:
5086:
5063:
5048:
5024:
4985:further reading
4957:
4906:
4889:article should
4875:
4851:
4701:
4699:nth derivatives
4664:
4636:
4630:
4618:
4612:
4605:
4599:
4591:
4585:
4570:
4553:
4550:
4549:
4491:
4490:
4464:
4377:
4217:
4195:
4164:
4163:
4160:never even used
4138:
4133:
4132:
4085:
4004:
4003:
3949:
3927:
3926:
3922:
3907:
3896:
3840:
3839:
3740:
3721:
3664:
3663:
3627:
3617:
3599:
3598:
3538:
3537:
3462:
3461:
3355:
3354:
3316:
3303:
3275:
3262:
3200:
3181:
3171:
3170:
3139:
3126:
3109:
3096:
3079:
3052:
3035:
3016:
3009:
3008:
2942:
2941:
2894:
2888:
2877:
2871:
2833:
2832:
2803:
2797:
2786:
2780:
2746:
2745:
2719:
2701:
2695:
2684:
2678:
2663:
2640:
2639:
2613:
2595:
2572:
2571:
2547:
2529:
2511:
2501:
2500:
2447:
2409:
2403:
2391:
2345:
2290:
2275:
2241:
2222:
2207:
2190:
2180:
2173:
2172:
2139:
2101:
2095:
2083:
2037:
1982:
1967:
1933:
1914:
1899:
1882:
1872:
1865:
1864:
1787:
1729:
1728:
1652:
1594:
1593:
1530:
1529:
1473:
1467:
1451:
1450:
1414:
1391:
1365:
1355:
1349:
1337:
1326:
1322:
1310:
1287:
1281:
1265:
1264:
1257:
1256:
1227:
1209:
1115:
1114:
1081:
1014:
1013:
983:
942:
941:
884:
883:
785:
780:
779:
773:
745:
723:
722:
701:
700:
679:
678:
657:
656:
655:(and of course
629:
628:
601:
600:
564:
541:
536:
535:
504:
484:
483:
480:
443:
440:
428:
418:
377:
369:
316:
308:
255:
251:
241:
234:
233:
211:
193:
172:
131:
128:
125:
122:
121:
99:
92:
72:
43:on Knowledge's
40:
30:
12:
11:
5:
6020:
6018:
6010:
6009:
6004:
5999:
5994:
5989:
5984:
5979:
5974:
5964:
5963:
5915:
5912:
5911:
5910:
5872:
5869:
5857:188.25.181.147
5819:
5816:
5800:
5797:
5783:
5781:
5780:
5761:
5755:
5751:
5747:
5741:
5736:
5733:
5732:
5725:
5721:
5717:
5712:
5703:
5699:
5695:
5690:
5685:
5682:
5681:
5677:
5672:
5668:
5664:
5659:
5652:
5649:
5645:
5636:
5632:
5628:
5623:
5618:
5615:
5614:
5608:
5605:
5600:
5597:
5589:
5586:
5581:
5578:
5577:
5573:
5569:
5566:
5563:
5559:
5552:
5549:
5545:
5540:
5537:
5536:
5532:
5527:
5523:
5519:
5515:
5512:
5508:
5501:
5498:
5494:
5489:
5488:
5474:
5473:
5459:
5456:
5451:
5448:
5440:
5437:
5432:
5428:
5424:
5421:
5418:
5414:
5407:
5404:
5400:
5377:
5376:
5361:
5357:
5353:
5348:
5343:
5339:
5334:
5330:
5326:
5321:
5314:
5311:
5307:
5284:
5283:
5267:
5262:
5257:
5253:
5248:
5243:
5239:
5234:
5230:
5226:
5222:
5219:
5215:
5208:
5205:
5201:
5185:
5184:
5169:
5165:
5159:
5155:
5151:
5144:
5140:
5135:
5131:
5127:
5123:
5120:
5116:
5109:
5106:
5102:
5085:
5082:
5047:
5044:
5023:
5020:
5019:
5018:
5017:
5016:
5015:
5014:
4999:
4972:
4971:
4970:
4969:
4958:Sławomir Biały
4945:
4944:
4928:
4927:
4907:Sławomir Biały
4898:indiscriminate
4874:
4871:
4850:
4847:
4832:
4831:
4830:
4829:
4786:it should be:
4784:
4783:
4782:
4752:
4751:
4748:
4741:
4700:
4697:
4696:
4695:
4694:
4693:
4692:
4691:
4672:
4671:
4670:
4669:
4660:
4635:
4617:
4604:
4590:
4552:
4539:
4538:
4504:
4501:
4498:
4482:Sławomir Biały
4470:
4469:
4460:
4415:
4414:
4413:
4412:
4411:
4410:
4409:
4408:
4407:
4406:
4385:
4384:
4383:
4382:
4373:
4367:
4366:
4365:
4362:
4351:
4293:
4290:
4279:
4276:
4272:
4262:
4261:
4260:
4259:
4218:Sławomir Biały
4208:
4207:
4196:Sławomir Biały
4177:
4174:
4171:
4145:
4141:
4114:
4113:
4100:
4099:
4098:
4097:
4086:Sławomir Biały
4074:
4073:
4053:
4050:
4049:
4048:
4032:
4028:
4024:
4020:
4017:
4014:
4011:
3986:
3985:
3984:
3968:
3964:
3961:
3956:
3952:
3948:
3945:
3941:
3937:
3934:
3920:
3905:
3894:
3880:
3879:
3878:
3862:
3858:
3854:
3850:
3847:
3834:
3833:
3830:
3802:
3801:
3773:
3772:
3759:
3755:
3748:
3744:
3738:
3735:
3732:
3729:
3725:
3718:
3715:
3712:
3709:
3706:
3703:
3700:
3697:
3694:
3691:
3687:
3683:
3680:
3677:
3674:
3671:
3660:
3659:
3646:
3642:
3635:
3631:
3625:
3621:
3614:
3611:
3607:
3595:
3594:
3581:
3578:
3575:
3572:
3569:
3566:
3563:
3560:
3557:
3554:
3551:
3548:
3545:
3520:
3519:
3506:
3502:
3498:
3495:
3492:
3489:
3485:
3481:
3478:
3475:
3472:
3469:
3458:
3457:
3444:
3440:
3436:
3433:
3430:
3427:
3423:
3419:
3416:
3413:
3410:
3406:
3402:
3399:
3396:
3392:
3388:
3385:
3382:
3379:
3376:
3373:
3370:
3366:
3362:
3351:
3350:
3335:
3331:
3324:
3320:
3314:
3310:
3306:
3300:
3297:
3294:
3290:
3283:
3279:
3273:
3269:
3265:
3259:
3256:
3253:
3249:
3245:
3242:
3239:
3236:
3232:
3228:
3225:
3222:
3219:
3215:
3208:
3204:
3198:
3195:
3192:
3189:
3185:
3178:
3164:
3163:
3147:
3143:
3137:
3133:
3129:
3123:
3117:
3113:
3107:
3103:
3099:
3093:
3087:
3083:
3077:
3073:
3069:
3066:
3063:
3059:
3055:
3049:
3043:
3039:
3033:
3030:
3027:
3024:
3020:
3005:
3004:
2991:
2987:
2983:
2980:
2977:
2973:
2969:
2966:
2963:
2960:
2957:
2954:
2950:
2935:
2934:
2930:
2929:
2916:
2912:
2901:
2897:
2892:
2884:
2880:
2875:
2868:
2865:
2860:
2857:
2854:
2850:
2846:
2843:
2840:
2830:
2810:
2806:
2801:
2793:
2789:
2784:
2777:
2774:
2769:
2765:
2761:
2758:
2754:
2743:
2726:
2722:
2717:
2708:
2704:
2699:
2691:
2687:
2682:
2675:
2670:
2666:
2661:
2655:
2652:
2648:
2637:
2620:
2616:
2611:
2607:
2602:
2598:
2593:
2587:
2584:
2580:
2569:
2554:
2550:
2545:
2541:
2536:
2532:
2527:
2523:
2518:
2514:
2509:
2474:
2473:
2459:
2453:
2450:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2406:
2398:
2394:
2390:
2387:
2384:
2380:
2376:
2370:
2367:
2362:
2359:
2356:
2353:
2349:
2343:
2340:
2337:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2297:
2293:
2289:
2286:
2283:
2279:
2272:
2266:
2263:
2258:
2255:
2252:
2249:
2245:
2239:
2236:
2229:
2225:
2221:
2218:
2215:
2211:
2204:
2198:
2194:
2188:
2184:
2166:
2165:
2151:
2145:
2142:
2137:
2134:
2131:
2128:
2125:
2122:
2119:
2116:
2113:
2110:
2107:
2104:
2098:
2090:
2086:
2082:
2079:
2076:
2072:
2068:
2062:
2059:
2054:
2051:
2048:
2045:
2041:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1989:
1985:
1981:
1978:
1975:
1971:
1964:
1958:
1955:
1950:
1947:
1944:
1941:
1937:
1931:
1928:
1921:
1917:
1913:
1910:
1907:
1903:
1896:
1890:
1886:
1880:
1876:
1854:
1853:
1840:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1794:
1790:
1786:
1783:
1780:
1776:
1772:
1769:
1766:
1761:
1758:
1755:
1752:
1748:
1744:
1741:
1737:
1719:
1718:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1681:
1678:
1675:
1672:
1669:
1666:
1659:
1655:
1651:
1648:
1645:
1641:
1637:
1634:
1631:
1626:
1623:
1620:
1617:
1613:
1609:
1606:
1602:
1584:
1583:
1570:
1567:
1562:
1559:
1556:
1553:
1549:
1545:
1542:
1538:
1526:
1525:
1497:
1496:
1480:
1476:
1471:
1465:
1460:
1455:
1447:
1441:
1437:
1433:
1429:
1425:
1422:
1418:
1412:
1408:
1403:
1399:
1395:
1388:
1384:
1379:
1373:
1369:
1363:
1359:
1352:
1345:
1341:
1335:
1329:
1325:
1318:
1314:
1308:
1302:
1294:
1290:
1285:
1279:
1274:
1269:
1250:
1249:
1234:
1230:
1225:
1221:
1216:
1212:
1208:
1204:
1200:
1197:
1194:
1190:
1186:
1183:
1179:
1175:
1171:
1167:
1164:
1161:
1157:
1153:
1150:
1147:
1144:
1140:
1136:
1133:
1130:
1126:
1122:
1095:
1090:
1085:
1080:
1077:
1073:
1070:
1067:
1064:
1060:
1057:
1054:
1051:
1048:
1044:
1040:
1036:
1032:
1029:
1026:
1022:
997:
992:
987:
982:
979:
975:
972:
968:
965:
962:
958:
954:
950:
938:
937:
924:
921:
916:
913:
910:
907:
903:
899:
896:
892:
880:
879:
872:
856:
849:
848:
834:
831:
828:
825:
822:
819:
814:
811:
808:
805:
802:
799:
795:
791:
788:
772:
769:
731:
709:
687:
665:
643:
639:
636:
615:
611:
608:
597:
596:
582:
577:
574:
571:
567:
563:
560:
557:
554:
551:
547:
544:
526:natural number
511:
507:
503:
500:
497:
494:
491:
479:
476:
439:
436:
413:
412:
401:
398:
395:
390:
386:
383:
380:
376:
372:
366:
363:
360:
356:
352:
349:
346:
343:
340:
337:
334:
329:
325:
322:
319:
315:
311:
305:
302:
299:
295:
291:
288:
285:
282:
279:
276:
273:
270:
262:
258:
254:
248:
244:
210:
207:
204:
203:
176:
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:
6019:
6008:
6005:
6003:
6000:
5998:
5995:
5993:
5990:
5988:
5985:
5983:
5980:
5978:
5975:
5973:
5970:
5969:
5967:
5960:
5958:
5954:
5950:
5944:
5942:
5938:
5934:
5921:
5913:
5909:
5905:
5901:
5897:
5896:Deacon Vorbis
5893:
5889:
5888:
5887:
5886:
5882:
5878:
5870:
5868:
5866:
5862:
5858:
5854:
5846:
5845:
5841:
5840:
5836:
5835:
5831:
5830:
5826:
5823:
5817:
5815:
5814:
5810:
5806:
5805:129.138.32.84
5798:
5796:
5795:
5791:
5787:
5759:
5749:
5739:
5734:
5719:
5710:
5697:
5688:
5683:
5675:
5666:
5657:
5650:
5647:
5643:
5630:
5621:
5616:
5606:
5603:
5598:
5595:
5587:
5584:
5579:
5571:
5567:
5564:
5561:
5557:
5550:
5547:
5543:
5538:
5530:
5521:
5513:
5510:
5506:
5499:
5496:
5492:
5479:
5478:
5477:
5457:
5454:
5449:
5446:
5438:
5435:
5430:
5426:
5422:
5419:
5416:
5412:
5405:
5402:
5398:
5389:
5388:
5387:
5385:
5380:
5355:
5346:
5341:
5337:
5328:
5319:
5312:
5309:
5305:
5296:
5295:
5294:
5292:
5289:According to
5287:
5265:
5255:
5246:
5241:
5237:
5228:
5220:
5217:
5213:
5206:
5203:
5199:
5190:
5189:
5188:
5167:
5163:
5153:
5142:
5138:
5129:
5121:
5118:
5114:
5107:
5104:
5100:
5091:
5090:
5089:
5081:
5079:
5075:
5071:
5067:
5060:
5057:
5054:
5051:
5043:
5042:
5038:
5034:
5030:
5027:
5021:
5013:
5009:
5005:
5000:
4998:
4994:
4990:
4986:
4982:
4978:
4977:
4976:
4975:
4974:
4973:
4968:
4964:
4960:
4953:
4949:
4948:
4947:
4946:
4943:
4939:
4935:
4930:
4929:
4925:
4920:
4919:
4918:
4917:
4913:
4909:
4903:
4899:
4894:
4892:
4888:
4884:
4880:
4872:
4870:
4869:
4865:
4861:
4857:
4848:
4846:
4845:
4841:
4837:
4827:
4823:
4819:
4815:
4811:
4807:
4803:
4799:
4795:
4792:
4788:
4787:
4785:
4780:
4776:
4772:
4768:
4764:
4760:
4756:
4755:
4754:
4753:
4749:
4746:
4742:
4739:
4735:
4731:
4727:
4724:
4723:
4722:
4721:Furthermore:
4719:
4718:
4714:
4710:
4706:
4698:
4690:
4686:
4682:
4678:
4677:
4676:
4675:
4674:
4673:
4668:
4663:
4658:
4654:
4650:
4646:
4640:
4634:
4628:
4622:
4616:
4609:
4603:
4595:
4589:
4582:
4578:
4574:
4567:
4563:
4557:
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4430:
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2018:
2012:
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1987:
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1976:
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1962:
1956:
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1948:
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201:
200:its talk page
196:
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5929:14 June 2024
5917:
5874:
5851:— Preceding
5847:
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5070:83.250.91.84
5064:— Preceding
5061:
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4625:instead of d
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4241:User:Maschen
4237:this version
4159:
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3989:
3917:
3913:
3909:
3902:
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1721:right-hand d
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419:
415:
225:is given by
220:
214:
212:
179:
148:Mid-priority
147:
107:
73:Mid‑priority
51:WikiProjects
34:
5877:Bapi paul07
5002:course...--
4987:section. --
4891:be restored
4808:, positive
4791:real number
4569:instead of
1586:left-hand d
524:, for some
444:—Preceding
195:its history
123:Mathematics
114:mathematics
70:Mathematics
41:Start-class
5966:Categories
4981:Referances
4705:WP:MOSMATH
4271:functions.
3908:)=0 ) and
429:Talk to me
223:chain rule
184:page were
5949:Not Fidel
5937:Not Fidel
5818:Languages
5384:This page
5291:this page
4934:Anuclanus
4879:this edit
4794:constants
4651:, where d
4233:this edit
4106:Thenub314
3522:whatever
851:because:
752:Philmac88
420:D O N D E
213:I merged
39:is rated
5853:unsigned
5066:unsigned
4856:See also
4810:integers
4275:section.
4249:Jowa fan
4158:that is
4079:There's
760:contribs
748:unsigned
743:etc.).
446:unsigned
424:groovily
5004:Maschen
4989:Maschen
4932:etc).--
4860:Maschen
4836:Maschen
4828:, ...".
4781:, ...",
4709:Maschen
4681:Maschen
4657:Quondum
4528:Maschen
4457:Quondum
4438:Maschen
4387:Fine.
4370:Quondum
4118:Maschen
4081:WP:TLDR
4063:Maschen
4058:clarify
3787:Maschen
1512:w.r.t.
465:Maschen
150:on the
5904:videos
5900:carbon
4824:, and
4584:, and
4575:= cos
4571:d sin
4564:= cos
4453:excess
4307:) or (
4213:WP:WPM
3897:(i.e.
186:merged
47:scale.
5786:Liron
4902:WP:OR
4662:contr
4462:contr
4375:contr
2488:and d
188:into
28:This
5953:talk
5927:and
5894:? –
5881:talk
5861:talk
5809:talk
5790:talk
5074:talk
5037:talk
5008:talk
4993:talk
4979:The
4963:talk
4938:talk
4912:talk
4864:talk
4840:talk
4796:are
4789:"...
4777:and
4736:and
4728:and
4713:talk
4685:talk
4560:sin
4532:talk
4507:blah
4442:talk
4357:vs.
4299:and
4253:talk
4223:talk
4201:talk
4180:blah
4122:talk
4110:talk
4091:talk
4067:talk
4000:)):
3910:then
3825:and
3817:and
3809:and
3791:talk
756:talk
469:talk
454:talk
5943:).
4524:why
4474:all
4326:or
4247:.
3780:all
3526:or
2379:lim
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2244:lim
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2040:lim
1970:lim
1936:lim
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1775:lim
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1640:lim
1612:lim
1548:lim
902:lim
863:, d
142:Mid
5968::
5955:)
5906:)
5902:•
5883:)
5863:)
5811:)
5792:)
5784:--
5565:
5562:ln
5514:
5511:ln
5420:
5417:ln
5386::
5293::
5221:
5218:ln
5122:
5119:ln
5076:)
5039:)
5010:)
4995:)
4965:)
4940:)
4914:)
4866:)
4842:)
4820:,
4816:,
4804:,
4800:,
4769:,
4765:,
4761:,
4715:)
4687:)
4629:=
4598:=
4534:)
4444:)
4425:.
4359:f′
4319:,
4317:fx
4255:)
4225:)
4215:.
4203:)
4124:)
4093:)
4069:)
4027:→
4013:∘
3963:→
3944:−
3857:→
3793:)
3696:∫
3670:∫
3494:∫
3477:∫
3432:∫
3415:∫
3384:∫
3361:∫
3299:∫
3258:∫
3241:∫
3224:∫
3177:∫
2859:∫
2853:ℓ
2845:∫
2757:ℓ
2651:ℓ
2583:ℓ
2513:ℓ
2480:/d
2449:δ
2432:−
2423:δ
2397:−
2389:→
2383:δ
2366:δ
2358:→
2352:δ
2333:δ
2330:−
2318:−
2296:−
2288:→
2282:δ
2262:δ
2254:→
2248:δ
2235:δ
2228:−
2220:→
2214:δ
2141:δ
2124:−
2115:δ
2081:→
2075:δ
2058:δ
2050:→
2044:δ
2019:−
2010:δ
1980:→
1974:δ
1954:δ
1946:→
1940:δ
1927:δ
1912:→
1906:δ
1830:δ
1827:−
1815:−
1793:−
1785:→
1779:δ
1765:δ
1757:→
1751:δ
1689:−
1680:δ
1650:→
1644:δ
1630:δ
1622:→
1616:δ
1566:δ
1558:→
1552:δ
1504:/d
1424:⋯
1398:⋯
1328:⋯
1185:⋯
1149:⋯
1066:⋯
1028:⋯
920:δ
912:→
906:δ
876:is
867:/d
762:)
758:•
721:,
699:,
677:,
638:∈
610:∈
573:−
471:)
456:)
365:π
362:∈
355:∏
351:⋅
321:π
304:Π
301:∈
298:π
294:∑
5951:(
5939:(
5898:(
5879:(
5859:(
5807:(
5788:(
5760:2
5754:|
5750:x
5746:|
5740:x
5735:=
5724:|
5720:x
5716:|
5711:x
5702:|
5698:x
5694:|
5689:1
5684:=
5676:)
5671:|
5667:x
5663:|
5658:(
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5580:=
5572:)
5568:u
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5531:)
5526:|
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5507:(
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5225:|
5214:(
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4763:B
4759:A
4747:.
4738:g
4734:f
4730:G
4726:F
4711:(
4683:(
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4176:=
4173:f
4170:d
4144:n
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4120:(
4112:)
4108:(
4089:(
4065:(
4031:R
4023:R
4019::
4016:g
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3994:g
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3960:}
3955:0
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3940:R
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3933:f
3921:0
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3886:(
3884:f
3861:R
3853:R
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3846:f
3827:g
3823:f
3819:g
3815:f
3811:g
3807:f
3789:(
3758:t
3754:d
3747:t
3743:d
3737:)
3734:t
3731:(
3728:x
3724:d
3717:)
3714:)
3711:t
3708:(
3705:x
3702:(
3699:y
3693:=
3690:x
3686:d
3682:)
3679:x
3676:(
3673:y
3645:t
3641:d
3634:t
3630:d
3624:x
3620:d
3613:=
3610:x
3606:d
3580:)
3577:t
3574:(
3571:x
3568:=
3565:x
3562:,
3559:)
3556:x
3553:(
3550:y
3547:=
3544:y
3528:q
3524:p
3505:p
3501:d
3497:q
3491:+
3488:q
3484:d
3480:p
3474:=
3471:q
3468:p
3443:p
3439:d
3435:q
3429:+
3426:q
3422:d
3418:p
3412:=
3409:p
3405:d
3401:q
3398:+
3395:q
3391:d
3387:p
3381:=
3378:)
3375:q
3372:p
3369:(
3365:d
3334:t
3330:d
3323:t
3319:d
3313:p
3309:d
3305:q
3296:+
3293:t
3289:d
3282:t
3278:d
3272:q
3268:d
3264:p
3255:=
3252:p
3248:d
3244:q
3238:+
3235:q
3231:d
3227:p
3221:=
3218:t
3214:d
3207:t
3203:d
3197:)
3194:q
3191:p
3188:(
3184:d
3146:t
3142:d
3136:p
3132:d
3128:q
3122:+
3116:t
3112:d
3106:q
3102:d
3098:p
3092:=
3086:t
3082:d
3076:p
3072:d
3068:q
3065:+
3062:q
3058:d
3054:p
3048:=
3042:t
3038:d
3032:)
3029:q
3026:p
3023:(
3019:d
2990:p
2986:d
2982:q
2979:+
2976:q
2972:d
2968:p
2965:=
2962:)
2959:q
2956:p
2953:(
2949:d
2915:x
2911:d
2900:2
2896:x
2891:d
2883:2
2879:y
2874:d
2867:+
2864:1
2856:=
2849:d
2842:=
2839:L
2809:2
2805:x
2800:d
2792:2
2788:y
2783:d
2776:+
2773:1
2768:x
2764:d
2760:=
2753:d
2725:2
2721:y
2716:d
2707:2
2703:x
2698:d
2690:2
2686:x
2681:d
2674:+
2669:2
2665:x
2660:d
2654:=
2647:d
2619:2
2615:y
2610:d
2606:+
2601:2
2597:x
2592:d
2586:=
2579:d
2553:2
2549:y
2544:d
2540:+
2535:2
2531:x
2526:d
2522:=
2517:2
2508:d
2494:L
2490:y
2486:y
2482:x
2478:y
2458:]
2452:x
2444:)
2441:x
2438:(
2435:y
2429:)
2426:x
2420:+
2417:x
2414:(
2411:y
2405:[
2393:0
2386:x
2375:=
2369:x
2361:0
2355:x
2342:]
2339:)
2336:x
2327:x
2324:(
2321:y
2315:)
2312:x
2309:(
2306:y
2303:[
2292:0
2285:x
2271:=
2265:x
2257:0
2251:x
2238:y
2224:0
2217:y
2203:=
2197:x
2193:d
2187:y
2183:d
2150:]
2144:x
2136:)
2133:x
2130:(
2127:y
2121:)
2118:x
2112:+
2109:x
2106:(
2103:y
2097:[
2089:+
2085:0
2078:x
2067:=
2061:x
2053:0
2047:x
2034:]
2031:)
2028:x
2025:(
2022:y
2016:)
2013:x
2007:+
2004:x
2001:(
1998:y
1995:[
1988:+
1984:0
1977:x
1963:=
1957:x
1949:0
1943:x
1930:y
1920:+
1916:0
1909:y
1895:=
1889:x
1885:d
1879:y
1875:d
1858:y
1839:]
1836:)
1833:x
1824:x
1821:(
1818:y
1812:)
1809:x
1806:(
1803:y
1800:[
1789:0
1782:x
1771:=
1768:y
1760:0
1754:y
1743:=
1740:y
1736:d
1723:y
1704:]
1701:)
1698:x
1695:(
1692:y
1686:)
1683:x
1677:+
1674:x
1671:(
1668:y
1665:[
1658:+
1654:0
1647:x
1636:=
1633:y
1625:0
1619:y
1608:=
1605:y
1601:d
1588:y
1569:x
1561:0
1555:x
1544:=
1541:x
1537:d
1524::
1522:x
1518:y
1514:x
1510:y
1506:x
1502:y
1479:n
1475:x
1470:d
1464:y
1459:n
1454:d
1446:=
1440:x
1436:d
1432:x
1428:d
1421:x
1417:d
1411:y
1407:d
1402:d
1394:d
1387:=
1383:)
1378:)
1372:x
1368:d
1362:y
1358:d
1351:(
1344:x
1340:d
1334:d
1324:(
1317:x
1313:d
1307:d
1301:=
1293:n
1289:x
1284:d
1278:y
1273:n
1268:d
1233:n
1229:x
1224:d
1220:=
1215:n
1211:)
1207:x
1203:d
1199:(
1196:=
1193:x
1189:d
1182:x
1178:d
1174:x
1170:d
1166:=
1163:)
1160:x
1156:d
1152:(
1146:)
1143:x
1139:d
1135:(
1132:)
1129:x
1125:d
1121:(
1094:x
1089:n
1084:d
1079:=
1076:x
1072:d
1069:d
1063:d
1059:=
1056:)
1053:)
1050:)
1047:x
1043:d
1039:(
1035:d
1031:(
1025:(
1021:d
996:x
991:2
986:d
981:=
978:x
974:d
971:d
967:=
964:)
961:x
957:d
953:(
949:d
923:x
915:0
909:x
898:=
895:x
891:d
869:x
865:y
861:y
859:d
833:d
830:i
827:p
824:u
821:t
818:s
813:g
810:n
807:i
804:f
801:f
798:e
794:=
790:′
787:f
754:(
730:Z
708:Q
686:N
664:R
642:C
635:n
614:N
607:n
595:"
581:.
576:1
570:n
566:x
562:n
559:=
556:)
553:x
550:(
546:′
543:f
529:n
510:n
506:x
502:=
499:)
496:x
493:(
490:f
467:(
452:(
400:)
397:x
394:(
389:)
385:|
382:B
379:|
375:(
371:g
359:B
348:)
345:)
342:x
339:(
336:g
333:(
328:)
324:|
318:|
314:(
310:f
290:=
287:)
284:)
281:x
278:(
275:g
272:(
269:f
261:n
257:x
253:d
247:n
243:d
202:.
154:.
53::
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