140:, and other geometrical notions that, although related to the subject of the article are not required to understand what is a derivative, and are of no help to understand, for example, that the velocity is the derivative of the position. Moreover, the article appears to be based on the strange idea that, understanding a notion is easier if the definition is explained before to be given. My opinion is that understanding a definition is impossible if this definition is not given before the explanation. This article is also based on another strange idea that not using the common terminology helps understanding (replacing "limit" by "limiting value", for example). As it is presently, the article is, maybe, useful as auxiliary textbook for US students of elementary courses of mathematics, but it is confusing for all other readers, and in particular for all the readers that are looking for accurate information.
3920:
an unpleasant look daemon hovering nearby. The daemon is like a person, but has wings, is less than 4 feet tall, and stays in the air like a humming bird. Upon meeting your gaze, the daemon grins, waves his hand, and two massive copies of the great wall of China appear. One of the two walls obstructs your view of the road ahead. The other wall obstructs your view the road already traveled. You can now only see the parts of the road a scant 10 feet in front and behind. The daemon now asks you a question: "Of the parts of the road you can see with your eyes, are there any sharp bends, or sudden turns?" If you answer honestly, and the answer is no, then the road you are walking along is differentiable where you currently stand.
350:. The proper definition of the reals need the notion of limit (or the equivalent notion of least upper bound). The definition of a continuous function involves the notion of limit. Thus in the standard logical model of mathematics (Zermelo-Fraenkel, with axiom of choice) and with the standard definition of of the reals as the smallest complete ordered field containing the rationals, there is a unique definition of the derivative. It may be possible to define it without using limits, but limits must be replaced by another notion which is not really simpler (least upper bound or shadow of a non standard real number). In any case mentioning such approaches in the beginning of the article is confusing for most audience. This, and
194:
what the derivative is good for. My point of view is that the article should not make that assumption. The article should instead assume that the reader does not know what a derivative is and does not know why a derivative is useful. This is why the article builds slowly to the definition: It is attempting to explain to the reader what kind of information is stored in derivative and why the definition of the derivative is the right one to capture that information. It is probable that the article could be better at this. It is possible that rearranging the article so that the definition came first and the justification came later would be an improvement, but I'm doubtful.
1451:
sentence from a detached point of view, heck even from a grammatical point of view: it contains the word "change" 3 times, the word "average" three times, and the word "function" twice. It is a mess, and, sadly, it is the cornerstone of the article because that sentence purports to offer the core definition of "derivative". I have a strong feeling that if you had an expert writer craft it, one who is gifted in the art of words, that the concept of derivative could be defined in a much more user friendly manner. But I've said my piece. I'm out.
85:
must contain an open interval containing the point of derivation) and at line 61 for the definition itself. Other readers may come to this article to understand some related terminology, such "derivative with respect to a variable" which is used in the lead, and apparently not defined in this article; presently the article is of few help for them. The definition of the derivative is preceded by explanations which, before the definition, may be useful only for people using WP as a textbook (but
1430:
years. Calculus was discovered relatively lately, and the reason, I think, is because it is an intrinsically difficult subject. It poses difficult problems philosophically, mathematically, and computationally. Don't be dismayed if you find calculus difficult; that just means you need to continue working at it, same as everyone. But even though it is difficult, it is surmountable. You can understand calculus! It will take effort, but it can be done, and it is very rewarding.
31:
1126:
f(a))..." a is still a point, and f(a) is the same as 'graph of f" at a. While you can write (a, f(a)) like coordinates, it only means f(a) around a. Yet now the slope is different that 'graph of f' at a (though close.) Continuing "...and a nearby point on the graph..." A nearby point on the graph? That treats (a, f(a)) as coordinates. Con't "... for example (a + h, f(a + h))." Sure enough, f(a+h) is a coordinate.
284:. There's an obvious one-dimensional subspace of this vector space, namely the span of any non-constant affine function; functions with classical derivatives are those that determine elements of this subspace. (The vector space isn't one dimensional, because it has classes for functions like the absolute value function. Therefore you get some non-classical derivatives this way. I think that for
1255:, and background knowledge (which probably has, as its primary determinants, the content and level of formal education or intensive self-directed study, and experience from work and "hobbies"). Its not unreasonable to argue that an encyclopedia is not a textbook, and can't reasonably hope to serve the needs of, say, someone who wants to understand differentiation but hasn't previously learned
3190:
3638:
of another curve of y =(x+dx) at a point where we seize x or y=x deliberately and introduce delta x OR when function y = f(x) changed to y=f(x+Δx). The value of x has reached to its maximum value instead of unlimited when a curve y=x doesn’t continue anymore at a point where we introduce delta x or dx as y=x and y =(x+dx) are two different types of curve (two diffrent functions).
2239:
4235:. When a person embarks on a journey to learn about the limit and differentiation their prior knowledge should include some familiarity with the concepts of the function, and the domain of a function. It is reasonable for contributors to an article to assume certain prior knowledge and not aim to insert all necessary prior knowledge into the article.
3630:
AC also increases and ultimately SECANT when x increases. Our goal is to bring dx, dy and AC to zero (not away from zero either positively or negatively - Point C has to be on the curve) or secant to tangent by reducing them close to zero but here dx heads toward zero but dy and AC increases when x increases on axis mathematically.
1950:"The derivative is a fundamental tool of calculus for studying the behavior of functions of a real variable." This doesn't explicitly define what a derivative is. The definition should be something alone the lines of "A derivative of a function is the instantaneous rate of change of the function with respect to its variable(s)
4326:
also changes. I do not have faith on negative values, we may oblige by norms of limits within an ordinal by doing reduction to its count, that's how the negative limits works out in real calculations, and this is how, I hope, as per my reading on
Mathematics Today magazine of MTG Group considering Limits.—
3645:
Similarly, dy is the small vertical change in y, therefore, we take the sum of all the small vertical lengths not the whole slice or y-coordinate(s) from zero to its value on the curve when we integrate both sides of the equation of dy = 2xdx but it turns into function of x or area under the graph –
3633:
Although the difference in the length of dx and dy can be noticeable clearly on the graph if we examine the triangle ABC at two different points for a gradient (dy/dx), say when an angle BAC = 0.1 degrees and 89.9 degrees on the curve but UNIT CIRCLE is the best example for observing the change in an
1511:
I would add that the reason why the sentence above contains "change" three times, "average" three times, and "function" twice is because it is precise. When the words in the sentence are interpreted correctly, then the sentence is a faithful rendering into words of the mathematical definition of the
1420:
I challenge ANY OF YOU to find 10 people with full time technical careers (or 100 people off the street) and I doubt even one person will know what that sentence is trying to say without having to read it multiple times. After reading the first paragraph of this article, I am going elsewhere because
1391:
I have tagged this section as disputed because it is self-contradictory: in the first paragraph, the fluxion and the dot notation denote an infinitesimal quantity, while in the last paragraph they denote, as usually, the derivative. Moreover, the first paragraph (including the example) is unclear and
1357:
Correct. Encyclopedias are books full of information that can only be understood by people that already know that information. They are designed to be read by academics with at least a bachelor's degree in that particular subfield for preparation. Oh wait, that's not what an encyclopedia is at all.
1288:
the other night that the only institutions that have survived the last 500 years unchanged in their essential nature are universities. So the inherent structure of knowledge, rather than the tendency of privilege to be used to preserve privilege, is probably the explanation for the academic system --
84:
Many readers of this article either come from another article through a wikilink of simply need to be recalled of the exact definition. The present state of the article is highly confusing for them: The definition of the derivative appear only at line 56 for the hypothesis (the domain of the function
3629:
The proportionality of both the angle CAB and dy with x are in contradiction with the proportionality of x and dx in the triangle ABC after probing the equation of dy/dx = 2x beyond its derivation on a graph of y = x2. When x increases; dx decreases, dy increases, and angle CAB increases. This means
1450:
I didn't want a book on calculus, only to understand the concept of a derivative, in plain language. I am questioning the "target audience" on this article, which seems like it is directed towards experts. Encyclopedias are meant to be informative, right? Consider your target audience. Read that
1125:
Currently, text is "The derivative of y with respect to x at a is, geometrically, the slope of the tangent line to the graph of f at a..." This presents 'graph of f' as a curve, which exists at point a. Continuing "...The slope of the tangent line is very close to the slope of the line through (a,
372:
Since functions determine measures, the Radon–Nikodym derivative of a measure determines the Radon–Nikodym derivative of a function. While it is strictly more general, there are no difference quotients involved. One surely needs limits of sequences somewhere in that, but that's a different concept
193:
However, I think our biggest disagreement is about the purpose of the article. As I see it, you view the article as a reference for information about the derivative. Please correct me if I'm wrong on this, but I think you are making the implicit assumption that the reader has some understanding of
179:
I think that the notions of graph and slope will be familiar to this article's audience. The audience may be familiar with secant and tangent lines, but maybe not. I would expect them to be least familiar with limits. I might be wrong. My view is limited mostly to
America and American education,
3919:
Imagine that you are walking along a long and winding road. After awhile, you stop shuffling your feet, and simply stand and look around. The road can be seen stretching far off into the distance; eventually disappearing on the horizon. You now notice a humming noise. You look to your right to find
3637:
RISE = dy = 2x and RUN = dx = 1 (always constant) in a GRADIENT of 1 in 2x which we obtained from the Eq1 of dy/dx=2x /1 at any point on the curve when there is no difference between secant and tangent – No idea how do we get dy/dx = 2x.dx but above said contradiction may be due to the introduction
207:
I am not sure that limits are necessary to define the derivative of a real function of a real variable. For instance, the Radon–Nikodym theorem does not need limits, only open sets, and it proves the existence of derivatives. Or one could take a
Zariski tangent space approach as follows: Consider
4325:
As per
Mathematics, in a ratio, considering the denominator value, when we do multiplication or a division, the denominator only increases. But in physics we have laws like directly proportional and in-directly proportional, so we quantify the value and go for the next calculation and thereby unit
4138:
In mathematics, if the domain of a function is not what can be determined by inspection that domain must be specified explicitly. If the domain is not specified explicitly, the reader should assume it can be determined accurately by inspection. Looking at your formula, there is no specification of
3554:
Pick any two points A and B close to each other on the curve of y =x. The coordinates of A on the curve are (x, y) or (x, x). Add Δx at A as usual. When x increases by Δx, then y increases by Δy. The x changes from x to (x +Δx) while y changes from y to (y + Δy) or f(x) to (x+Δx). Thus the x and y
2406:
Some students may be misled by the nice animated figure of the difference quotient into thinking that the formal definition of the derivative is given in terms of limits from the right, in which case the existence of the derivative of f at x_0 would not imply continuity at x_0, but only continuity
1987:
1429:
Euclid once said, "There is no royal road to geometry". It is the same way with calculus; calculus is intrinsically a very difficult subject. Much of the mathematics that is taught before calculus has been known in some form for five hundred years or more; some topics date back over two thousand
1411:
I am trying to learn derivatives. I consider myself a pretty smart person because I am a fluent programmer in Visual Studio. However, I have never needed calculus in life, hence I completely forgot it. So here I come to this article hoping to learn about the logic of a derivative. Instead I am
154:
Another remark: The heading of the sections are confusing. The subject of section "Differentiation and the derivative" (without its subsections) is "Geometrical interpretation" and the subject of the subsection "Rigourous definition" is "Geometric interpretation of the definition". It cannot be a
104:
I disagree with your view of the article. I do not think that many readers of this article come here to recall the exact definition. Instead I think they come here because they do not understand the concept of the derivative. This is why the article starts slowly, building up motivation for the
1881:
When we take some sample derivatives d/dx(x^2) is 2x and d/dx(x^3) is 3x^2. It means that differentiation gives the mirroring capability. To maintain atomicity and to avoid any internal transformation or any collateral damage dy = dx must be maintained. All the objects whichever has life over the
4230:
I don’t agree that we are talking about a technical convention. In the explanation of any topic within any subject (or the teaching of that topic) there is a logical progression from the elementary to the introductory to the core and ultimately on to an advanced understanding. Most of what came
3641:
Further, integration is the reverse process of differentiation. Although delta x or dx is ignored during the process of derivation of dy/dx becaue of their small values but we can’t ignore them in the process of integration which makes a lot of difference in summation. They can’t be disappeared
1475:
That is exactly the problem: plain language is not convenient to describe mathematical concepts. This is the reason for which mathematicians have introduced many words and new meanings for older words. Even that is not sufficient; therefore the introduction of formulas. Personally, I would have
1914:
So dy = dx general equality must be examined in any sector of science. Particularly in
Telecom. I mean Atomicity of Living Organisms must be considered. If it is not also Transformation Process must be speculated with R^2 Norms. Birth ratio cannot end in extinction as it can be seen that it is
1553:
Please state clearly where you think that the article is wrong. About the existence of non-differentiable continuous functions, the article gives two examples of continuous functions that are not differentiable at the origin (the absolute value and the cubic root function) and one example of a
368:
I agree that all alternatives to limits require concepts that are at least as complicated. I am coming around to your viewpoint that the beginning of the article should discuss the standard definition of the derivative, and other definitions, if they are mentioned at all, should be near the
2843:
Initially select a function. Then quantise it and don't draw a graph based exactly on the function, but instead create a near match constituted of many of its derivatives. Then find an algorithm which finds almost perfect matches of derivative fragmentations which simulate the function.
1435:
I recommend reading a textbook. Because
Knowledge is structured like an encyclopedia, it is not especially pedagogical, and therefore it's very difficult to learn mathematics from Knowledge. You will find a textbook easier to learn from; they're written to be learned from, after all.
3923:
That is the nature of what a function means to be differentiable. A road is differentiable everywhere, if, no matter you are on the road, the daemon can limit your field view so that no sharp bends, or turns are visible on the piece of road just in front of you, and just behind you.
2918:
When condition
Collinear not met, then the matter is within R^3, meaning Euclidean Space, by dividing the possibility in to TWO, just like how in any triangle, by a line (derivative) there can be two right angled triangle formed and all else angles,- the ratio is precise.
180:
and it's colored by my own preference for geometry. I do agree that the geometric viewpoint taken in the article will not be much help to readers who are more inclined to, say, physics. Newton's motivation was physical, so perhaps the article should discuss some physics.
1416:"The derivative measures the instantaneous rate of change of the function, as distinct from its average rate of change, and is defined as the limit of the average rate of change in the function as the length of the interval on which the average is computed tends to zero."
1512:
derivative. If you compare the sentence and the definition closely, then you might be able to see the similarities. Mathematics is inherently a logical discipline, and we rely on both equations and formal language to communicate that logic accurately to the reader.
3550:
Derivatives show us how fast something is changing at any point. For example; the gradient of the graph of y = x at any point is twice the value of x thereat. The process of finding the derivation of a gradient / slope of a function y=f(x) or y = x is as follow.
3581:
ABC is an infinitesimal triangle made by dx, dy, and hypotenuse or slope of tangent where point A and C are always on the curve. Length of AB = Base = dx, Length of BC = Perpendicular= dy and Length of
Hypotenuse = AC. Angle CAB or BAC is the slope of a tangent
4238:
The IP User is exploring our article on differentiation without an adequate familiarity of the concept of the domain of a function. That problem will be easily rectified when they read my reply and look at the article about the domain to which I blue linked.
3933:
in the
Knowledge article formula is restricted to be positive. That means that you would be able to look at the road ahead, but never look behind you. In that case, the absolute value of x is differentiable at zero, which is not correct. If were standing at
151:. As no correct definition of the derivative may exist without these notions, it is definitely impossible to understand the concept of derivative without understanding these two notions. Therefore the removed section starts as slowly as reasonably possible.
1129:
The problem is that when the writer says 'slope at a,' a is an x,y point. When he writes (a, f(a)), it first continues to refer to a as an x,y point, but later a seems to become only a point on the x axis. How else to interpret '(a + h, f(a + h))'?
3720:
Maybe it is pragmatic, so tried all least possibilities that they shouldn't have done possibly, as per necessity, it would have been done or carried out. Frankly speaking we should give up. So we could feel flying not the pragmatic things.
1309:
Thanks for that.. I actually did check out the
Wikibooks links to the various textbooks, but they weren't much help either.. But I understand what you're saying.. maybe if I actually bothered to pay attention to high school math lol! --
3035:
is that, however there as,- one stays as lock, the ordinal, there exists a continuous changing ordinal with it the whole thing is precise by. And it is completely another division of mathematics, other than the way, you says about
2234:{\displaystyle {\frac {d}{dx}}\int _{\alpha (x)}^{\beta (x)}f(x,t)dt=f\left(x,\beta \left(x\right)\right){\frac {d\beta }{dx}}-f(x,\alpha (x)){\frac {d\alpha }{dx}}+\int _{\alpha (x)}^{\beta (x)}{\frac {\partial f}{\partial x}}dt}
2749:
may define either square, but may be ambiguous only when the domain and the codomain are equal, and a multiplication is defined on the codomain. In this case, the notation should be clarified or avoided. Also, do not confuse
296:. But I don't know how one would prove that.) I don't think these should be covered in the article, but I do think they justify my belief that the intuitive concept of a derivative does not necessarily depend on limits.
373:
than the limit of a function. It is therefore an approach to derivatives which does not involve limits of functions. Since the Radon–Nikodym theorem is important, I have added a link to it in the "See also" section.
4207:
So, I agree with the IP user that the article must be edited. However, fixing the issue would require to rewrite completely the article. As, presently, I have not the time for that, I'll simply tag the article.
4339:
There is no such thing called by negative limits, rather it's so called, left sided or left-hand limit and right sided or right-hand limit. The way you mentioned, I hope you're referring to left sided limit.
3649:
The derivation of the natural relationship of a gradient of 1 in 2x at any point with y=x or dy/dx=2x is still unbeknownst to illuminates - Anyone who is in agreement with all above unless satisfied by logic.
1496:
In this sentence, I have linked the words which have an accurate mathematical meaning which is not exactly the same as their usual meaning. I am not sure which sentence is easier to understand for the layman.
1241:(with sub-projects that each have a respective major subject area as another parent) to review the lead sections of the more technical articles woth the goal that as large as feasible a fraction of users will
1424:
I'm not saying to explain it in dumbed down language, but is it too much to ask that when you present advanced topics like this that you can assume the reader is not already a scholar in the said subject?
310:
Radon-Nikodym derivative is not the derivative of a function, but the derivative of a measure. Thus this is not an alternative definition, but a generalization. Your "Zariski tangent space approach" seems
1728:
3393:
3303:
767:
3117:
2854:
2417:
2716:
376:
Yes, the Zariski tangent space approach is OR. I hope I did not give any other impression. I am not proposing that we include it in the article. But I do not see why it should be true that
4041:. It is very important you be able to look a little ways behind yourself, as well as ahead of yourself. I really hate that formula which was provided, because the formula makes it looks like
354:
are the obvious reasons to remove the mention of generalizations and other approaches of the derivatives in the beginning of the article. I have already removed them and I will do it again.
2410:
Anyone who clicks the link on limits will be saved from that misapprehension, but a well-placed word on that point in the caption of the figure might save someone from needless confusion.
1817:
Take y=f(x)=constant, then x=f(y)=δ(y) [an axis change leads to a dirac delta on y=x and this is not differenciable due the discontinuity, therefore y=f(x)=constant is not differenciable.
1251:
seen those in the last 7 years -- rather than bcz of their existing under other names), the ability to tackle a given article depends importantly on at least two factors about the reader:
155:
rigourous definition (what is a non-rigourous definition?) because of the number of geometric terms that do not appear in any correct definition (graph, tangent line, secant line, slope).
4110:
3905:
2600:
4165:
You are right, but the reader is not supposed to be aware of these technical conventions. Presently the article is polluted by lengthy explanations of why one considers derivative (
1882:
earth is collinear on sphere (Earth). I mean δ = 0. Forget about the death. Its natural disaster. But Consider any signals on & around us. it supposed to obey dy = dx equality.
1331:
The purpose of this article isn't to teach calculus, but to describe what it. This description may involve complex vocabulary and mathematics; it's not meant to pander to beginners
2642:
2544:
3067:
2977:
1263:, but going around for an extended period w/o being able to grasp why it was worthwhile to single out that concept for a formal definition; that memory leaves me suspecting that
3695:
Maybe this can be tried with x . y rather, however so it is not,- "neither f(x) nor Y". And about f(x,y) I can't talk about it at all, as per me it is completely hypothetical —
886:
812:
4013:
2822:
637:
1774:
541:
3412:
Not done: This would be correct, but is unneeded, as the reader is not supposed to know what is a difference quotient; knowing it is not helpful for reading the formula.
1001:
949:
586:
4198:
3618:
2460:
2775:
2747:
1071:
1028:
917:
839:
664:
508:
3003:
4039:
3984:
3958:
2348:
2344:
2330:
969:
3527:
Y' touches Y for which axis x quantity reduced by x' amount. And always, there could be a derivative up-to the situation, till f' (f dash) cannot be obtained 😉
2436:
The article uses notation like g^2 to denote x↦(g(x))^2. While this is common notation in some branches, it is also common in other branches to read this as
1209:
needed for understanding a particular article in this WP. Rather, its purpose is to serve non-native speakers of English whose English vocabulary is small,
1268:
anyone who first encounters the concept of "function" in WP will need a textbook -- if not an instructor -- in addition to WP, to understand any plausible
1421:
this article has no value to me, it doesn't aim to teach to an outsider, it only aims to lecture to its own students at best, and worst, its own peers.
2847:
Why to do this? In some machines it's easier to control binarily the graph of many derivative fragments (limited lines) than the original function.
3166:
If feels ugly to me to have the function subtracted from the limit, and it doesn't matter which one you subtract from since it's an absolute value.
1259:(if i correctly recall the title of what i once took). I recall being convinced (during the course that i mean) that i had absorbed the concept of
4265:
where we consider the one-sided limits. The derivative is only defined if the left derivative and right derivatives exist and have equal value. --
2259:
110:
Because of this I've removed your new section. I'm still open to improvements in the article, but I don't think the new section is a good idea.
4290:. This includes moving some subsections as true sections. IMO, this fixes the ambiguity pointed out by the IP user. I have also removed the tag
1087:
I don't remember where I saw this. It's possibly an exercise in one of Karl Stromberg's textbooks on analysis, but I don't have a copy handy.
1220:
128:
I disagree with your assertion that the article starts slowly: To understand correctly the first sentences, one has to understand what is the
3655:
3614:
3398:
3317:
3227:
3167:
1824:
1621:
1538:
100 books put example of funtion continous but no differenciable, to absolute valor, please use other example, for this case: y = |1-|x||,--
1140:
2979:
is well defined and commonly cosidered in mathematics. So an encyclopedia, and specifically Knowledge, must not be restricted to the case
4115:
2901:
1458:
1365:
3121:
2858:
2421:
1238:
1180:
2326:
When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
2316:
4286:
I have added a formal definition that includes the definition of the involved limit, and renamed the section "Differentiation" as
89:) and are confusing for the others. For these reasons I'll add immediately after the lead a section "Definition and terminology".
3651:
3610:
3493:
1641:
3110:
In the "generalizations" section, there should be information about "functional derivatives" and the "calculus of variations"
2477:
1931:
4015:... then the daemon could limit your view of the road ahead of you such that you cannot see the bend in the road located at
1094:
1037:
1915:
slightly reduced with the count 7 out 10 in its ability level. Science on Higgs principles must take this consideration.
1885:
In 2D if the condition is Collinear means, its a straight line. Lets take the Straight line form with Slope m and assess.
3432:
Ladies, Gentlemen, please let me say you that we can define derivative without infinity's or limit's notion, as follows:
1223:. I'm not sure we have any distinct articles with this role, and i'm pretty sure there is no template along the lines of
2391:
669:
3742:
I do not know much around f(x,y). But I know one among the problem comes under the section un-computability, so called
1412:
overwhelmed with technical jargon that only someone who already is fluent in calculus has any hope of understanding.
1965:
I agree with you. I edited the first sentence so it know describes a derivative as the instantaneous rate of change.
2647:
4294:
3005:
However, the mention of Euclidean space is irrelevant here, as the Euclidean structure is not used. I'll fix this.
38:
2492:: Could you indicate where this notation is used in this article? All exponents that I have found are exponent of
143:
On the other hand, in the section that you have removed, the only technical notions that appear are the notion of
3171:
1828:
1625:
288:
the vector space should be two dimensional, one dimension each for the positive and negative directions, and for
64:
59:
2347:
to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
1088:
1031:
105:
definition, and treating carefully the difference between the derivative at a point and the derivative function.
4173:
it is difficult to distinguish the true definition from intuitive explanations. For knowing that the increment
3402:
3086:
1864:
1155:
I agree that the article was not completely specific about the tangent line. Is the current revision better?
1144:
4053:
3848:
3162:
Can we change the part inside the limit of the definition of a derivative from |L - /h|<ε to |/h - L|<ε?
2549:
458:
by homogeneity). I am not sure how to prove or disprove this because I don't see any way to find elements in
4119:
1369:
1275:
anyone who first encounters "derivative" here will need a textbook or instructor to understand any plausible
3196:
3031:
is more than enough and definitely you'll be with me. And as the way you stated from, what I feel, in about
2897:
2382:
2308:
1462:
394:. Certainly this is false for a polynomial ring (though we are not in that situation). I think it that on
4345:
3821:
3634:
angle CAB (say 0.1 and 89.9 degrees) of a triangle ABC for dy and dx and the comparison of their lengths.
3535:
2282:
1227:
971:
in some open set only if the function actually is itself affine. (There is a way out of this by replacing
4287:
4262:
4170:
3555:
coordinates of B on the curve are (x + Δx, y + Δy) or (. Now the instantaneous rate of change is given by
3032:
1904:
That means x must & should be INTACT with its multiples of INCLINATION. In other norms it should be NP
1798:
1491:
1487:
1483:
1260:
4133:
of this function is all values of h except zero. "All values" includes both negative and positive values.
3037:
2884:
is generally applicable here. If you think normal spacing is that horrible, it should be changed at the
4166:
2880:
Incessant fiddling with spacing to make stuff appear more to your own preference is never appropriate.
2473:
2366:
If you have discovered URLs which were erroneously considered dead by the bot, you can report them with
2354:
86:
4310:, since these explanations, although convenient for a textbook, seem too detailed for an encyclopedia.
3136:
My guess is that this article requires citations in order to correspond with sources. In addition, why
3081:
can evolve with new patterns. And computers are there to make any such algorithms, I tried once with 90
2875:
2307:. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit
1578:
Having a function f(x) and 1st derivative ∂f(x), the integral can be calculated using the formulation:
4341:
4327:
3816:
3531:
3090:
2930:
2605:
2507:
2278:
1473:"I didn't want a book on calculus, only to understand the concept of a derivative, in plain language."
4270:
4130:
3785:
3606:
3489:
3152:
3113:
3043:
2953:
2850:
2465:
2413:
1919:
1820:
1617:
1555:
1454:
1361:
1136:
3722:
3696:
844:
772:
4249:
4201:
4149:
3312:
3144:
2489:
2469:
1970:
1479:
1316:
1193:
4139:
the domain so the reader should not entertain any apprehension that it excludes negative numbers.
3989:
4315:
4213:
3417:
3010:
2893:
2828:
2780:
1927:
1563:
1543:
1502:
1397:
1276:
359:
162:
94:
2351:
before doing mass systematic removals. This message is updated dynamically through the template
2317:
https://web.archive.org/web/20060415161115/http://www.its.caltech.edu/~sean/book/unabridged.html
591:
3646:
no idea how but summation of vertical lengths on a graph gives vertical length only not curve?
3601:
The length of dx = dy when Angle CAB = 45 degrees The length of dx < dy when Angle CAB : -->
2367:
1737:
889:
513:
4304:
2248:
1795:
974:
922:
546:
4180:
4169:), and the definition of the derivative is delayed after these explanations. Even in section
2439:
3773:
3070:
3024:
2923:
1955:
1593:
Yeah, this is original verifiable derivation, original research I suppose, peer reviewable.
1336:
1256:
2374:
1810:
The reason for all differenciation to stop when the differenciation has reached a constant.
4300:, as explanations do not need to be mathematically accurate. However, I have left the tag
4266:
3485:
3148:
2948:
2753:
2725:
1909:
1296:
1252:
1049:
1006:
895:
817:
642:
486:
47:
17:
2982:
4018:
3963:
3960:, then the road ahead would look like a straight line. If you were standing just before
3937:
2320:
1734:(from the chain rule) but you cannot move the derivative outside the integral, even for
4240:
4140:
3800:
3082:
2333:, "External links modified" talk page sections are no longer generated or monitored by
2267:
1966:
1878:
It means all the points on the same line. So surface of Triangle ∆ cannot be obtained.
1848:
1781:
1517:
1441:
1311:
1188:
1160:
1078:
472:
301:
115:
3910:
I am hoping that someone will make explicit that you must consider negative values of
2373:
If you found an error with any archives or the URLs themselves, you can fix them with
1179:
baffles me! Still it would be nice if someone could dumb it down enough to write up a
954:
4311:
4209:
3812:
3796:
3413:
3006:
2926:, as per me all are same, and it's just one Ordinal, say Constant or so, I mean R^n.
2887:
2824:
1923:
1559:
1539:
1498:
1393:
1184:
355:
351:
158:
90:
3023:
Thanks for making attention to the inquiry. About Numbers, I feel theories given by
3069:
is well defined and commonly considered in mathematics as the way you says in. So,
2881:
2244:
1285:
1202:
312:
3588:• dy/dx is directly proportional to x or angle CAB is directly proportional to x.
841:(despite the fact that this would be begging the question anyway). In that case
3666:
3500:
2340:
1951:
1814:
There is no further differenciation when the derivative has reached a constant.
1332:
46:
If you wish to start a new discussion or revive an old one, please do so on the
4349:
4334:
4319:
4274:
4256:
4217:
4156:
4123:
3823:
3737:
3711:
3688:
3659:
3622:
3539:
3522:
3421:
3406:
3175:
3156:
3125:
3097:
3014:
2937:
2905:
2862:
2832:
2481:
2425:
2396:
2286:
2271:
2258:
I agree that this is a useful formula. It appears (with different notation) at
2252:
1974:
1959:
1935:
1852:
1832:
1801:
1785:
1629:
1567:
1547:
1521:
1506:
1466:
1445:
1401:
1373:
1340:
1318:
1299:
1195:
1164:
1148:
1098:
1082:
1041:
476:
363:
305:
166:
119:
98:
3808:
3200:
3137:
3028:
2339:. No special action is required regarding these talk page notices, other than
2300:
1292:
1269:
3642:
forever and should resurface during the process of integration or summation.
3477:, derivative function becomes constant that is called Derivative of f(x) at x
3804:
3792:
3777:
3769:
3078:
2263:
1865:
http://trythissolution.blogspot.in/2014/01/how-to-find-area-of-triangle.html
1844:
1840:
1839:
If y=f(x)=constant, then there is no way to write x=f(y) because f is not a
1777:
1513:
1437:
1156:
1074:
468:
297:
111:
1213:
appropriate to the corresponding topic in any of the other languages of WP.
1046:
Beautiful! I'm amazed. Do you have a reference for your statements about
3591:• dx is indirectly proportional to x OR x is inversely proportional to dx
2943:
I cannot understand your phrasing. Nevertheless, for any positive integer
1245:
which links they need to follow to get up to speed for the article itself.
3765:
3141:
1175:
The language in this article still baffles me. In all honesty, calculus
3788:
3751:
3074:
3842:
The following formula is in the Knowledge article on derivatives:
2262:. Do you think there is a good place to mention it in this article?
2892:
template level, and not in the many thousands of individual uses. –
3761:
4129:
The formula you have quoted is a function of the variable h. The
3780:
like both z = f(x,y) and z(x,y) = the_same_FUNCTION, that's the
3568:
Reduce Δx close to zero by taking limit (Δx to dx and Δy to dy)
3465:, after equally eliminating denominator, is also true for x = x
3427:
3750:
is a finite set of unordered pairs of strings." - this is the
3184:
25:
3499:
That's just another way of saying that you take the limit.
3428:
Derivative's definition without infinity's or limit's notion
3781:
2311:
for additional information. I made the following changes:
1872:|x2 y2 1| = ∆ = 0 means that the condition is Collinear.
1723:{\displaystyle \int f(x)f'(x)\,dx={\frac {1}{2}}f(x)^{2},}
814:. Remarkably, this approach is not fixed by going up to
3388:{\displaystyle L=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}}
3298:{\displaystyle L=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}}
2915:
As part of this Article, only R^3 should be mentioned.
1221:
Minimal concepts for understanding differential calculus
315:
and appear to be completely buggy: I cannot imagine for
3665:
This has been answered at the Science Reference Desk.
2462:. The article should point this out where it's used.
2304:
324:
another definition than the set of functions such that
4167:
this makes sense in a textbook, not in an encyclopedia
1211:
while presuming the same level of underlying knowledge
4200:
can be negative, the reader has to go to the article
4183:
4056:
4021:
3992:
3966:
3940:
3851:
3320:
3230:
3046:
2985:
2956:
2783:
2756:
2728:
2650:
2608:
2552:
2510:
2442:
2321:
http://www.its.caltech.edu/~sean/book/unabridged.html
1990:
1910:
Talk:Derivative#The_problem_of_differential_equations
1740:
1644:
1554:
continuous function which is nowhere differentiable (
1052:
1009:
977:
957:
925:
898:
847:
820:
775:
672:
645:
594:
549:
516:
489:
292:
there should be one basis vector for each element of
3438:: For every real function f(x), derivative function
3085:
scribbling in G++. Any it's out, however it's under
1843:. In particular it is not a Dirac delta function.
2343:using the archive tool instructions below. Editors
762:{\displaystyle f=|f|^{1/2}|f|^{-1/2}f\in m_{a}^{2}}
4192:
4104:
4033:
4007:
3978:
3952:
3899:
3387:
3297:
3061:
2997:
2971:
2816:
2769:
2741:
2710:
2636:
2594:
2538:
2454:
2233:
1768:
1722:
1392:needs more explanations for being understandable.
1065:
1022:
995:
963:
943:
911:
880:
833:
806:
761:
658:
631:
580:
535:
502:
3811:circuit, that is what I could understand out of
3328:
3238:
1205:is intended for users are not prepared with the
3181:Semi-protected edit request on 14 November 2022
2711:{\displaystyle x\mapsto (g\circ g)(x)=g(g(x)).}
1482:of the ratio of the variation (or change) of a
2922:No matter what do you do with,- n, R^n, 1 and
2329:This message was posted before February 2018.
1494:on which the ratio is computed tends to zero."
3594:• dy is directly proportional to x.dx or dx
3585:According to the aforementioned Eq1 or Eq2-
1476:written the sentence that you have quoted as
888:is much too large (because of the failure of
8:
3784:. I am saying this because I couldn't find
2911:Euclidean R^3 should only referred, not R^n
892:), and you can conclude that a function in
3604:
3111:
2848:
2463:
2411:
2299:I have just modified one external link on
1605:∂x^a = a*x^(a-1)=a*x^a * x^(-1)=a*(x^a)/x
4182:
4105:{\displaystyle {\frac {f(a+h)-f(a)}{h}}.}
4057:
4055:
4020:
3991:
3965:
3939:
3900:{\displaystyle {\frac {f(a+h)-f(a)}{h}}.}
3852:
3850:
3343:
3331:
3319:
3253:
3241:
3229:
3073:is been clearly examined. So in this way
3053:
3048:
3045:
2984:
2963:
2958:
2955:
2788:
2782:
2761:
2755:
2733:
2727:
2649:
2619:
2607:
2595:{\displaystyle x\mapsto g(x).\cdot g(x),}
2551:
2530:
2509:
2441:
2402:Limits and figure illustrating derivative
2205:
2190:
2176:
2149:
2099:
2024:
2010:
1991:
1989:
1760:
1739:
1711:
1688:
1643:
1057:
1051:
1014:
1008:
987:
982:
976:
956:
935:
930:
924:
903:
897:
872:
867:
858:
852:
846:
825:
819:
798:
785:
780:
774:
753:
748:
728:
721:
716:
707:
697:
693:
688:
679:
671:
650:
644:
616:
609:
604:
595:
593:
568:
564:
559:
550:
548:
527:
515:
494:
488:
220:) of continuous real-valued functions on
2722:is the name of a function, the notation
1599:∂a^x = a^x * ln(a) or ∫a^x = a^x /ln(a)
1289:whatever media theorists may speculate.)
1003:by its closure in a natural topology on
398:there should be an explicit isomorphism
2260:Differentiation under the integral sign
1677:
145:real-valued function of a real variable
3118:2600:6c44:e7f:1100:80de:361b:400c:7d47
2855:2A02:587:4115:6100:8144:5CEE:EA1E:63B0
2418:2601:646:4200:68A0:E058:43C6:86F0:6886
1859:Condition Collinear and The Derivative
1636:This is not correct. It's true that:
1239:WikiProject intellectual accessibility
44:Do not edit the contents of this page.
4261:There might be a case for mentioning
3757:Elements of the Theory of Computation
3040:or maybe I think confident about so,
2432:Unclear notation: powers of functions
7:
3754:from the book, ISBN 81-203-1016-0 -
1611:∫x^a = /a = x^(a)*x^(1)/a=x*x^(a)/a
1201:It is an easy mistake to think that
2500:. In any case, the standard use in
4184:
3834:Can Someone explicitly State that
2216:
2208:
919:is in the congruence class modulo
24:
3768:. We do have a reference page in
3599:dy when Angle CAB < 45 degrees
2637:{\displaystyle x\mapsto g^{2}(x)}
2539:{\displaystyle x\mapsto g(x)^{2}}
2303:. Please take a moment to review
1584:Corralary to goat´s formulation:
1215:The article you want would be on
1187:) .. if that's even possible. --
3188:
3062:{\displaystyle \mathbf {R} ^{n}}
3049:
2972:{\displaystyle \mathbf {R} ^{n}}
2959:
1614:Doesn´t get your ehummm, goat?
1219:WP, with a title like, perhaps,
29:
3916:, not just positive values.
3803:we make / could arrive to sort
3546:Is derivation of dy/dx correct?
951:of an affine function for each
881:{\displaystyle m_{a}/m_{a}^{2}}
807:{\displaystyle m_{a}^{2}=m_{a}}
4090:
4084:
4075:
4063:
3885:
3879:
3870:
3858:
3376:
3370:
3361:
3349:
3335:
3286:
3280:
3271:
3259:
3245:
2906:15:19, 30 September 2020 (UTC)
2795:
2789:
2702:
2699:
2693:
2687:
2678:
2672:
2669:
2657:
2654:
2631:
2625:
2612:
2586:
2580:
2568:
2562:
2556:
2527:
2520:
2514:
2200:
2194:
2186:
2180:
2146:
2143:
2137:
2125:
2054:
2042:
2034:
2028:
2020:
2014:
1750:
1744:
1708:
1701:
1674:
1668:
1657:
1651:
717:
708:
689:
680:
605:
596:
560:
551:
1:
4008:{\displaystyle x=-0.00000001}
3422:17:00, 14 November 2022 (UTC)
3407:15:11, 14 November 2022 (UTC)
3015:17:19, 20 November 2020 (UTC)
2938:16:54, 20 November 2020 (UTC)
2863:15:11, 22 December 2018 (UTC)
2833:14:31, 18 November 2018 (UTC)
2482:13:37, 18 November 2018 (UTC)
2426:00:49, 27 December 2017 (UTC)
2397:07:02, 9 September 2017 (UTC)
1960:19:22, 15 December 2013 (UTC)
1341:19:18, 15 December 2013 (UTC)
1099:15:44, 10 December 2013 (UTC)
4335:17:46, 25 October 2021 (UTC)
3824:18:02, 30 October 2023 (UTC)
3738:12:31, 5 November 2019 (UTC)
3689:19:50, 2 November 2018 (UTC)
3660:22:49, 12 October 2018 (UTC)
3623:08:38, 19 October 2018 (UTC)
3523:19:49, 2 November 2018 (UTC)
3176:07:42, 27 October 2022 (UTC)
2817:{\displaystyle g^{(2)}=g''.}
2287:12:52, 1 February 2017 (UTC)
1936:20:22, 25 October 2014 (UTC)
1853:14:33, 25 October 2014 (UTC)
1833:12:59, 24 October 2014 (UTC)
1247:OTOH (and more likely why i
1165:02:45, 30 January 2014 (UTC)
1149:23:04, 29 January 2014 (UTC)
1083:16:49, 7 December 2013 (UTC)
1042:13:34, 7 December 2013 (UTC)
477:16:28, 4 December 2013 (UTC)
364:08:58, 4 December 2013 (UTC)
306:06:23, 4 December 2013 (UTC)
167:16:17, 3 December 2013 (UTC)
120:02:42, 3 December 2013 (UTC)
99:16:51, 2 December 2013 (UTC)
80:Definition of the derivative
3652:Eclectic Eccentric Kamikaze
3611:Eclectic Eccentric Kamikaze
3215:to reactivate your request.
3203:has been answered. Set the
3157:11:12, 24 August 2022 (UTC)
1802:19:04, 21 August 2014 (UTC)
1786:02:36, 19 August 2014 (UTC)
1630:18:39, 18 August 2014 (UTC)
1319:15:19, 5 October 2010 (UTC)
632:{\displaystyle |f|^{-1/2}f}
438:to the function that sends
232:determines a maximal ideal
87:Knowledge is not a textbook
4368:
4350:09:29, 3 August 2023 (UTC)
3838:is allowed to be negative?
2360:(last update: 5 June 2024)
2296:Hello fellow Wikipedians,
1975:19:47, 24 March 2015 (UTC)
1769:{\displaystyle f(x)=x^{3}}
1590:Proof: ∂f(x) * ∫f(x) = ^2
1558:). This seems sufficient.
1300:20:02, 4 August 2010 (UTC)
1183:version of the article - (
536:{\displaystyle f\in m_{a}}
4320:11:03, 29 June 2021 (UTC)
4275:04:17, 29 June 2021 (UTC)
4257:23:01, 28 June 2021 (UTC)
4218:13:36, 28 June 2021 (UTC)
4157:10:41, 28 June 2021 (UTC)
4124:03:58, 28 June 2021 (UTC)
4047:is strictly positive.
3712:01:36, 5 April 2019 (UTC)
3574:dy /dx = 2x------Eq1 OR
3126:23:26, 15 June 2021 (UTC)
3098:18:14, 8 March 2021 (UTC)
1568:09:10, 10 July 2014 (UTC)
1522:13:38, 25 June 2014 (UTC)
1507:08:32, 25 June 2014 (UTC)
1467:06:48, 25 June 2014 (UTC)
1446:04:28, 25 June 2014 (UTC)
1272:encyclopedia article, and
1196:03:22, 4 April 2010 (UTC)
1121:Formal Definition unclear
996:{\displaystyle m_{a}^{2}}
944:{\displaystyle m_{a}^{2}}
581:{\displaystyle |f|^{1/2}}
4193:{\displaystyle \Delta x}
3760:distributed to me under
3087:Creative Commons license
2455:{\displaystyle g\circ g}
2272:01:24, 11 May 2011 (UTC)
2253:14:47, 10 May 2011 (UTC)
1548:18:25, 9 July 2014 (UTC)
1486:by the variation of its
1402:18:24, 4 June 2014 (UTC)
1374:03:06, 23 May 2014 (UTC)
1181:Simple English Knowledge
430:defined by (in the case
241:, and the derivative of
4233:assumed prior knowledge
3776:. If we could make the
3540:16:29, 3 May 2023 (UTC)
3494:14:11, 2 May 2018 (UTC)
2292:External links modified
1478:"the derivative is the
4263:Semi-differentiability
4194:
4171:§ Rigourous definition
4106:
4035:
4009:
3980:
3954:
3901:
3598:The length of dx : -->
3389:
3299:
3140:exclusively discusses
3063:
3033:Reentrancy (computing)
2999:
2973:
2818:
2771:
2743:
2712:
2638:
2596:
2540:
2456:
2235:
1770:
1724:
1596:You can verify using:
1261:function (mathematics)
1067:
1024:
997:
965:
945:
913:
882:
835:
808:
763:
660:
633:
582:
537:
504:
4195:
4107:
4036:
4010:
3986:... say, standing at
3981:
3955:
3902:
3799:, it's like over the
3795:, however I can find
3390:
3311:and the limit of the
3300:
3106:Functional Derivative
3064:
3000:
2974:
2819:
2772:
2770:{\displaystyle g^{2}}
2744:
2742:{\displaystyle g^{2}}
2713:
2639:
2597:
2541:
2457:
2236:
1771:
1725:
1284:(Someone remarked to
1279:encyclopedia article.
1068:
1066:{\displaystyle C^{1}}
1025:
1023:{\displaystyle C^{1}}
998:
966:
946:
914:
912:{\displaystyle m_{a}}
883:
836:
834:{\displaystyle C^{1}}
809:
764:
661:
659:{\displaystyle m_{a}}
634:
583:
538:
505:
503:{\displaystyle C^{0}}
454:||) (i.e., extending
42:of past discussions.
4181:
4054:
4019:
3990:
3964:
3938:
3849:
3318:
3228:
3044:
2998:{\displaystyle n=3.}
2983:
2954:
2781:
2754:
2726:
2648:
2606:
2550:
2508:
2504:mathematics is that
2440:
2341:regular verification
1988:
1738:
1642:
1556:Weierstrass function
1207:underlying knowledge
1050:
1007:
975:
955:
923:
896:
845:
818:
773:
670:
643:
592:
547:
514:
487:
4034:{\displaystyle x=0}
3979:{\displaystyle x=0}
3953:{\displaystyle x=0}
3313:difference quotient
3038:Euclidean structure
2331:After February 2018
2204:
2038:
1587:∂f(x)= ^2 / ∫f(x).
1387:Fluent and fluxions
992:
940:
877:
790:
758:
130:graph of a function
4190:
4102:
4031:
4005:
3976:
3950:
3897:
3385:
3342:
3295:
3252:
3059:
2995:
2969:
2814:
2767:
2739:
2708:
2634:
2592:
2536:
2452:
2407:from the right.
2385:InternetArchiveBot
2336:InternetArchiveBot
2231:
2172:
2006:
1897:Slope(m) = Δy/x
1766:
1720:
1678:
1581:∫f(x)=^2 / ∂f(x).
1574:Goat´s Formulation
1277:partial derivative
1063:
1020:
993:
978:
961:
941:
926:
909:
878:
863:
831:
804:
776:
759:
744:
656:
629:
578:
533:
500:
4295:confusing section
4097:
3892:
3625:
3609:comment added by
3577:dy = 2x.dx --Eq2
3571:dy /dx = 2x + dx
3383:
3327:
3293:
3237:
3219:
3218:
3128:
3116:comment added by
2924:Direction Cosines
2865:
2853:comment added by
2484:
2468:comment added by
2428:
2416:comment added by
2361:
2223:
2167:
2117:
2004:
1946:Initial sentence.
1939:
1922:comment added by
1823:comment added by
1696:
1620:comment added by
1457:comment added by
1364:comment added by
1185:Simple:Derivative
1139:comment added by
964:{\displaystyle a}
77:
76:
54:
53:
48:current talk page
4359:
4309:
4303:
4299:
4293:
4252:
4231:earlier must be
4199:
4197:
4196:
4191:
4176:
4152:
4111:
4109:
4108:
4103:
4098:
4093:
4058:
4046:
4040:
4038:
4037:
4032:
4014:
4012:
4011:
4006:
3985:
3983:
3982:
3977:
3959:
3957:
3956:
3951:
3932:
3915:
3906:
3904:
3903:
3898:
3893:
3888:
3853:
3819:
3774:Semi-Thue system
3686:
3565:Δy/ Δx = / 1
3562:Δy/ Δx = / Δx
3520:
3460:
3458:
3457:
3451:
3448:
3394:
3392:
3391:
3386:
3384:
3379:
3344:
3341:
3304:
3302:
3301:
3296:
3294:
3289:
3254:
3251:
3210:
3206:
3192:
3191:
3185:
3068:
3066:
3065:
3060:
3058:
3057:
3052:
3025:Pierre de Fermat
3004:
3002:
3001:
2996:
2978:
2976:
2975:
2970:
2968:
2967:
2962:
2946:
2891:
2879:
2823:
2821:
2820:
2815:
2810:
2799:
2798:
2776:
2774:
2773:
2768:
2766:
2765:
2748:
2746:
2745:
2740:
2738:
2737:
2721:
2717:
2715:
2714:
2709:
2643:
2641:
2640:
2635:
2624:
2623:
2601:
2599:
2598:
2593:
2545:
2543:
2542:
2537:
2535:
2534:
2461:
2459:
2458:
2453:
2395:
2386:
2359:
2358:
2337:
2240:
2238:
2237:
2232:
2224:
2222:
2214:
2206:
2203:
2189:
2168:
2166:
2158:
2150:
2118:
2116:
2108:
2100:
2098:
2094:
2093:
2037:
2023:
2005:
2003:
1992:
1980:a useful formula
1938:
1916:
1901:x.Slope(m) = Δy
1894:x = (y - C) / m
1835:
1790:Also, ∫x^a = /(a
1775:
1773:
1772:
1767:
1765:
1764:
1729:
1727:
1726:
1721:
1716:
1715:
1697:
1689:
1667:
1632:
1469:
1376:
1314:
1257:advanced algebra
1232:
1226:
1191:
1151:
1091:
1072:
1070:
1069:
1064:
1062:
1061:
1034:
1029:
1027:
1026:
1021:
1019:
1018:
1002:
1000:
999:
994:
991:
986:
970:
968:
967:
962:
950:
948:
947:
942:
939:
934:
918:
916:
915:
910:
908:
907:
890:Hadamard's lemma
887:
885:
884:
879:
876:
871:
862:
857:
856:
840:
838:
837:
832:
830:
829:
813:
811:
810:
805:
803:
802:
789:
784:
768:
766:
765:
760:
757:
752:
737:
736:
732:
720:
711:
706:
705:
701:
692:
683:
665:
663:
662:
657:
655:
654:
638:
636:
635:
630:
625:
624:
620:
608:
599:
587:
585:
584:
579:
577:
576:
572:
563:
554:
542:
540:
539:
534:
532:
531:
509:
507:
506:
501:
499:
498:
483:For elements of
429:
283:
263:
249:is the class of
138:slope of a curve
73:
56:
55:
33:
32:
26:
4367:
4366:
4362:
4361:
4360:
4358:
4357:
4356:
4307:
4301:
4297:
4291:
4250:
4179:
4178:
4174:
4150:
4059:
4052:
4051:
4042:
4017:
4016:
3988:
3987:
3962:
3961:
3936:
3935:
3928:
3911:
3854:
3847:
3846:
3840:
3817:
3786:Uncomputability
3764:as part of our
3667:
3548:
3501:
3482:
3480:
3476:
3468:
3464:
3456:
3452:
3449:
3446:
3442:
3441:
3439:
3430:
3399:131.175.177.193
3345:
3316:
3315:
3255:
3226:
3225:
3208:
3204:
3189:
3183:
3168:198.183.167.143
3164:
3138:History section
3134:
3108:
3071:Rolle's theorem
3047:
3042:
3041:
2981:
2980:
2957:
2952:
2951:
2949:Euclidean space
2944:
2913:
2885:
2873:
2871:
2841:
2803:
2784:
2779:
2778:
2757:
2752:
2751:
2729:
2724:
2723:
2719:
2646:
2645:
2615:
2604:
2603:
2548:
2547:
2526:
2506:
2505:
2438:
2437:
2434:
2404:
2389:
2384:
2352:
2345:have permission
2335:
2309:this simple FaQ
2294:
2215:
2207:
2159:
2151:
2109:
2101:
2083:
2073:
2069:
1996:
1986:
1985:
1982:
1948:
1917:
1907:
1861:
1825:201.208.189.225
1818:
1812:
1756:
1736:
1735:
1707:
1660:
1640:
1639:
1622:201.208.189.225
1615:
1576:
1536:
1452:
1409:
1389:
1359:
1312:
1253:cognitive style
1230:
1224:
1189:
1173:
1171:Simple English?
1134:
1123:
1089:
1053:
1048:
1047:
1032:
1010:
1005:
1004:
973:
972:
953:
952:
921:
920:
899:
894:
893:
848:
843:
842:
821:
816:
815:
794:
771:
770:
715:
687:
668:
667:
646:
641:
640:
603:
590:
589:
558:
545:
544:
523:
512:
511:
490:
485:
484:
466:
428:
419:
411:
393:
384:
349:
340:
323:
282:
273:
265:
250:
240:
82:
69:
30:
22:
21:
20:
18:Talk:Derivative
12:
11:
5:
4365:
4363:
4355:
4354:
4353:
4352:
4288:§ Explanations
4284:
4283:
4282:
4281:
4280:
4279:
4278:
4277:
4236:
4223:
4222:
4221:
4220:
4205:
4189:
4186:
4160:
4159:
4135:
4134:
4113:
4112:
4101:
4096:
4092:
4089:
4086:
4083:
4080:
4077:
4074:
4071:
4068:
4065:
4062:
4030:
4027:
4024:
4004:
4001:
3998:
3995:
3975:
3972:
3969:
3949:
3946:
3943:
3908:
3907:
3896:
3891:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3863:
3860:
3857:
3839:
3832:
3831:
3830:
3829:
3828:
3827:
3826:
3801:Timing diagram
3715:
3714:
3692:
3691:
3580:
3559:Δy/ Δx = /
3558:
3547:
3544:
3543:
3542:
3528:
3525:
3478:
3474:
3466:
3462:
3454:
3444:
3434:
3429:
3426:
3425:
3424:
3382:
3378:
3375:
3372:
3369:
3366:
3363:
3360:
3357:
3354:
3351:
3348:
3340:
3337:
3334:
3330:
3326:
3323:
3292:
3288:
3285:
3282:
3279:
3276:
3273:
3270:
3267:
3264:
3261:
3258:
3250:
3247:
3244:
3240:
3236:
3233:
3224:and the limit
3217:
3216:
3193:
3182:
3179:
3163:
3160:
3133:
3130:
3107:
3104:
3103:
3102:
3101:
3100:
3056:
3051:
3018:
3017:
2994:
2991:
2988:
2966:
2961:
2912:
2909:
2870:
2867:
2840:
2837:
2836:
2835:
2813:
2809:
2806:
2802:
2797:
2794:
2791:
2787:
2764:
2760:
2736:
2732:
2707:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2671:
2668:
2665:
2662:
2659:
2656:
2653:
2633:
2630:
2627:
2622:
2618:
2614:
2611:
2591:
2588:
2585:
2582:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2533:
2529:
2525:
2522:
2519:
2516:
2513:
2451:
2448:
2445:
2433:
2430:
2403:
2400:
2379:
2378:
2371:
2324:
2323:
2315:Added archive
2293:
2290:
2275:
2274:
2242:
2241:
2230:
2227:
2221:
2218:
2213:
2210:
2202:
2199:
2196:
2193:
2188:
2185:
2182:
2179:
2175:
2171:
2165:
2162:
2157:
2154:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2115:
2112:
2107:
2104:
2097:
2092:
2089:
2086:
2082:
2079:
2076:
2072:
2068:
2065:
2062:
2059:
2056:
2053:
2050:
2047:
2044:
2041:
2036:
2033:
2030:
2027:
2022:
2019:
2016:
2013:
2009:
2002:
1999:
1995:
1981:
1978:
1947:
1944:
1942:
1905:
1860:
1857:
1856:
1855:
1811:
1808:
1807:
1806:
1805:
1804:
1763:
1759:
1755:
1752:
1749:
1746:
1743:
1732:
1731:
1730:
1719:
1714:
1710:
1706:
1703:
1700:
1695:
1692:
1687:
1684:
1681:
1676:
1673:
1670:
1666:
1663:
1659:
1656:
1653:
1650:
1647:
1575:
1572:
1571:
1570:
1535:
1532:
1531:
1530:
1529:
1528:
1527:
1526:
1525:
1524:
1432:
1431:
1408:
1405:
1388:
1385:
1384:
1383:
1382:
1381:
1380:
1379:
1378:
1377:
1348:
1347:
1346:
1345:
1344:
1343:
1324:
1323:
1322:
1321:
1304:
1303:
1302:
1290:
1281:
1280:
1273:
1265:
1264:
1246:
1214:
1172:
1169:
1168:
1167:
1141:208.80.117.214
1122:
1119:
1118:
1117:
1116:
1115:
1114:
1113:
1112:
1111:
1110:
1109:
1108:
1107:
1106:
1105:
1104:
1103:
1102:
1101:
1090:Sławomir Biały
1060:
1056:
1033:Sławomir Biały
1017:
1013:
990:
985:
981:
960:
938:
933:
929:
906:
902:
875:
870:
866:
861:
855:
851:
828:
824:
801:
797:
793:
788:
783:
779:
756:
751:
747:
743:
740:
735:
731:
727:
724:
719:
714:
710:
704:
700:
696:
691:
686:
682:
678:
675:
653:
649:
628:
623:
619:
615:
612:
607:
602:
598:
575:
571:
567:
562:
557:
553:
530:
526:
522:
519:
497:
493:
462:
424:
415:
389:
380:
374:
370:
345:
336:
319:
278:
269:
236:
224:. Each point
200:
199:
198:
197:
196:
195:
186:
185:
184:
183:
182:
181:
172:
171:
170:
169:
156:
152:
141:
123:
122:
107:
106:
81:
78:
75:
74:
67:
62:
52:
51:
34:
23:
15:
14:
13:
10:
9:
6:
4:
3:
2:
4364:
4351:
4347:
4343:
4338:
4337:
4336:
4333:
4332:
4331:
4324:
4323:
4322:
4321:
4317:
4313:
4306:
4296:
4289:
4276:
4272:
4268:
4264:
4260:
4259:
4258:
4255:
4253:
4246:
4245:
4244:
4237:
4234:
4229:
4228:
4227:
4226:
4225:
4224:
4219:
4215:
4211:
4206:
4203:
4187:
4172:
4168:
4164:
4163:
4162:
4161:
4158:
4155:
4153:
4146:
4145:
4144:
4137:
4136:
4132:
4128:
4127:
4126:
4125:
4121:
4117:
4116:73.153.62.136
4099:
4094:
4087:
4081:
4078:
4072:
4069:
4066:
4060:
4050:
4049:
4048:
4045:
4028:
4025:
4022:
4002:
3999:
3996:
3993:
3973:
3970:
3967:
3947:
3944:
3941:
3931:
3927:Suppose that
3925:
3921:
3917:
3914:
3894:
3889:
3882:
3876:
3873:
3867:
3864:
3861:
3855:
3845:
3844:
3843:
3837:
3833:
3825:
3822:
3820:
3814:
3813:Uncomputation
3810:
3806:
3802:
3798:
3797:Uncomputation
3794:
3790:
3787:
3783:
3779:
3775:
3771:
3767:
3763:
3759:
3758:
3753:
3749:
3748:A Thue system
3745:
3741:
3740:
3739:
3736:
3734:
3731:
3728:
3727:
3719:
3718:
3717:
3716:
3713:
3710:
3708:
3705:
3702:
3701:
3694:
3693:
3690:
3687:
3685:
3682:
3679:
3676:
3673:
3670:
3664:
3663:
3662:
3661:
3657:
3653:
3647:
3643:
3639:
3635:
3631:
3627:
3626:
3624:
3620:
3616:
3612:
3608:
3600:
3595:
3592:
3589:
3586:
3583:
3578:
3575:
3572:
3569:
3566:
3563:
3560:
3556:
3552:
3545:
3541:
3537:
3533:
3529:
3526:
3524:
3521:
3519:
3516:
3513:
3510:
3507:
3504:
3498:
3497:
3496:
3495:
3491:
3487:
3472:
3437:
3433:
3423:
3419:
3415:
3411:
3410:
3409:
3408:
3404:
3400:
3396:
3380:
3373:
3367:
3364:
3358:
3355:
3352:
3346:
3338:
3332:
3324:
3321:
3314:
3309:
3306:
3290:
3283:
3277:
3274:
3268:
3265:
3262:
3256:
3248:
3242:
3234:
3231:
3222:
3214:
3211:parameter to
3202:
3198:
3194:
3187:
3186:
3180:
3178:
3177:
3173:
3169:
3161:
3159:
3158:
3154:
3150:
3146:
3143:
3139:
3131:
3129:
3127:
3123:
3119:
3115:
3105:
3099:
3096:
3095:
3094:
3088:
3084:
3080:
3076:
3072:
3054:
3039:
3034:
3030:
3026:
3022:
3021:
3020:
3019:
3016:
3012:
3008:
2992:
2989:
2986:
2964:
2950:
2942:
2941:
2940:
2939:
2936:
2935:
2934:
2927:
2925:
2920:
2916:
2910:
2908:
2907:
2903:
2899:
2895:
2894:Deacon Vorbis
2889:
2883:
2877:
2868:
2866:
2864:
2860:
2856:
2852:
2845:
2838:
2834:
2830:
2826:
2811:
2807:
2804:
2800:
2792:
2785:
2762:
2758:
2734:
2730:
2705:
2696:
2690:
2684:
2681:
2675:
2666:
2663:
2660:
2651:
2628:
2620:
2616:
2609:
2589:
2583:
2577:
2574:
2571:
2565:
2559:
2553:
2531:
2523:
2517:
2511:
2503:
2499:
2495:
2491:
2487:
2486:
2485:
2483:
2479:
2475:
2471:
2467:
2449:
2446:
2443:
2431:
2429:
2427:
2423:
2419:
2415:
2408:
2401:
2399:
2398:
2393:
2388:
2387:
2376:
2372:
2369:
2365:
2364:
2363:
2356:
2350:
2346:
2342:
2338:
2332:
2327:
2322:
2318:
2314:
2313:
2312:
2310:
2306:
2302:
2297:
2291:
2289:
2288:
2284:
2280:
2273:
2269:
2265:
2261:
2257:
2256:
2255:
2254:
2250:
2246:
2228:
2225:
2219:
2211:
2197:
2191:
2183:
2177:
2173:
2169:
2163:
2160:
2155:
2152:
2140:
2134:
2131:
2128:
2122:
2119:
2113:
2110:
2105:
2102:
2095:
2090:
2087:
2084:
2080:
2077:
2074:
2070:
2066:
2063:
2060:
2057:
2051:
2048:
2045:
2039:
2031:
2025:
2017:
2011:
2007:
2000:
1997:
1993:
1984:
1983:
1979:
1977:
1976:
1972:
1968:
1963:
1961:
1957:
1953:
1945:
1943:
1940:
1937:
1933:
1929:
1925:
1921:
1912:
1911:
1902:
1898:
1895:
1892:
1889:
1886:
1883:
1879:
1876:
1873:
1870:
1867:
1866:
1858:
1854:
1850:
1846:
1842:
1838:
1837:
1836:
1834:
1830:
1826:
1822:
1815:
1809:
1803:
1800:
1797:
1793:
1789:
1788:
1787:
1783:
1779:
1761:
1757:
1753:
1747:
1741:
1733:
1717:
1712:
1704:
1698:
1693:
1690:
1685:
1682:
1679:
1671:
1664:
1661:
1654:
1648:
1645:
1638:
1637:
1635:
1634:
1633:
1631:
1627:
1623:
1619:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1573:
1569:
1565:
1561:
1557:
1552:
1551:
1550:
1549:
1545:
1541:
1533:
1523:
1519:
1515:
1510:
1509:
1508:
1504:
1500:
1495:
1493:
1489:
1485:
1481:
1474:
1471:
1470:
1468:
1464:
1460:
1459:67.182.153.76
1456:
1449:
1448:
1447:
1443:
1439:
1434:
1433:
1428:
1427:
1426:
1422:
1418:
1417:
1413:
1406:
1404:
1403:
1399:
1395:
1386:
1375:
1371:
1367:
1366:104.32.136.26
1363:
1356:
1355:
1354:
1353:
1352:
1351:
1350:
1349:
1342:
1338:
1334:
1330:
1329:
1328:
1327:
1326:
1325:
1320:
1317:
1315:
1308:
1307:
1306:
1305:
1301:
1298:
1294:
1287:
1283:
1282:
1278:
1274:
1271:
1267:
1266:
1262:
1258:
1254:
1250:
1244:
1240:
1236:
1229:
1222:
1218:
1212:
1208:
1204:
1200:
1199:
1198:
1197:
1194:
1192:
1186:
1182:
1178:
1170:
1166:
1162:
1158:
1154:
1153:
1152:
1150:
1146:
1142:
1138:
1131:
1127:
1120:
1100:
1096:
1092:
1086:
1085:
1084:
1080:
1076:
1058:
1054:
1045:
1044:
1043:
1039:
1035:
1015:
1011:
988:
983:
979:
958:
936:
931:
927:
904:
900:
891:
873:
868:
864:
859:
853:
849:
826:
822:
799:
795:
791:
786:
781:
777:
754:
749:
745:
741:
738:
733:
729:
725:
722:
712:
702:
698:
694:
684:
676:
673:
651:
647:
626:
621:
617:
613:
610:
600:
573:
569:
565:
555:
528:
524:
520:
517:
495:
491:
482:
481:
480:
479:
478:
474:
470:
465:
461:
457:
453:
449:
445:
441:
437:
434:= 0) mapping
433:
427:
423:
418:
414:
409:
405:
401:
397:
392:
388:
383:
379:
375:
371:
367:
366:
365:
361:
357:
353:
348:
344:
339:
335:
331:
327:
322:
318:
314:
309:
308:
307:
303:
299:
295:
291:
287:
281:
277:
272:
268:
261:
257:
253:
248:
244:
239:
235:
231:
227:
223:
219:
215:
211:
206:
205:
204:
203:
202:
201:
192:
191:
190:
189:
188:
187:
178:
177:
176:
175:
174:
173:
168:
164:
160:
157:
153:
150:
146:
142:
139:
135:
131:
127:
126:
125:
124:
121:
117:
113:
109:
108:
103:
102:
101:
100:
96:
92:
88:
79:
72:
68:
66:
63:
61:
58:
57:
49:
45:
41:
40:
35:
28:
27:
19:
4330:BramStoker's
4329:
4328:
4285:
4247:
4242:
4241:
4232:
4147:
4142:
4141:
4114:
4043:
3929:
3926:
3922:
3918:
3912:
3909:
3841:
3835:
3772:for this as
3756:
3755:
3747:
3744:Thue Systems
3743:
3732:
3729:
3725:
3723:
3706:
3703:
3699:
3697:
3683:
3680:
3677:
3674:
3671:
3668:
3648:
3644:
3640:
3636:
3632:
3628:
3605:— Preceding
3603:
3597:
3596:
3593:
3590:
3587:
3584:
3579:
3576:
3573:
3570:
3567:
3564:
3561:
3557:
3553:
3549:
3517:
3514:
3511:
3508:
3505:
3502:
3483:
3470:
3435:
3431:
3397:
3310:
3307:
3223:
3220:
3212:
3197:edit request
3165:
3135:
3112:— Preceding
3109:
3093:BramStoker's
3092:
3091:
2933:BramStoker's
2932:
2931:
2928:
2921:
2917:
2914:
2882:MOS:STYLERET
2876:Joel Brennan
2872:
2849:— Preceding
2846:
2842:
2501:
2497:
2493:
2464:— Preceding
2435:
2412:— Preceding
2409:
2405:
2383:
2380:
2355:source check
2334:
2328:
2325:
2298:
2295:
2276:
2243:
1964:
1949:
1941:
1918:— Preceding
1913:
1903:
1899:
1896:
1893:
1890:
1887:
1884:
1880:
1877:
1874:
1871:
1868:
1862:
1819:— Preceding
1816:
1813:
1796:Arthur Rubin
1791:
1616:— Preceding
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1537:
1477:
1472:
1453:— Preceding
1423:
1419:
1415:
1414:
1410:
1390:
1360:— Preceding
1286:Charlie Rose
1248:
1242:
1234:
1228:prerequisite
1216:
1210:
1206:
1176:
1174:
1135:— Preceding
1132:
1128:
1124:
666:as well, so
463:
459:
455:
451:
447:
443:
439:
435:
431:
425:
421:
416:
412:
407:
403:
399:
395:
390:
386:
381:
377:
346:
342:
337:
333:
332:) = 0. Thus
329:
325:
320:
316:
293:
289:
285:
279:
275:
270:
266:
259:
255:
251:
246:
242:
237:
233:
229:
225:
221:
217:
213:
209:
148:
144:
137:
133:
129:
83:
70:
43:
37:
4342:BilkTheHulk
3818:BilkTheHulk
3532:BilkTheHulk
3473:: For x = x
3221:substitue:
3132:Reconstruct
3083:° (degrees)
2279:Diu Gaikwad
1891:y - C = mx
1888:y = mx + C
1490:, when the
36:This is an
4267:Salix alba
4003:0.00000001
3809:electronic
3602:45 degrees
3486:Georges T.
3471:Definition
3443:f(x) - f(x
3205:|answered=
3201:Derivative
3149:Dedhert.Jr
3029:Pythagoras
2392:Report bug
2301:Derivative
1908:.x Refer,
1875:|x3 y3 1|
1869:|x1 y1 1|
1407:GobblyGook
1270:derivative
1177:in general
3815:article.—
3805:bijection
3793:wikipedia
3778:bijection
3770:wikipedia
3735:Sadasivam
3709:Sadasivam
3484:Regards.
3461:for x ≠ x
3079:Chemistry
2839:Machinery
2498:functions
2494:variables
2375:this tool
2368:this tool
1967:Joeygrill
1841:bijection
1243:recognize
1133:BrianMC
208:the ring
71:Archive 3
65:Archive 2
60:Archive 1
4312:D.Lazard
4305:textbook
4210:D.Lazard
3766:syllabus
3619:contribs
3607:unsigned
3414:D.Lazard
3142:calculus
3114:unsigned
3007:D.Lazard
2851:unsigned
2825:D.Lazard
2490:77nafets
2478:contribs
2470:77nafets
2466:unsigned
2414:unsigned
2381:Cheers.—
1932:contribs
1924:Ansathas
1920:unsigned
1821:unsigned
1618:unsigned
1560:D.Lazard
1540:Peiffers
1499:D.Lazard
1492:interval
1488:variable
1484:function
1455:unsigned
1394:D.Lazard
1362:unsigned
1137:unsigned
356:D.Lazard
159:D.Lazard
91:D.Lazard
4243:Dolphin
4143:Dolphin
3789:article
3752:excerpt
3395:exists
3305:exists
3145:history
3075:Physics
2869:Spacing
2305:my edit
2245:Jackzhp
1863:Refer,
1249:haven't
1237:need a
639:are in
543:, then
147:and of
134:tangent
39:archive
4131:domain
3530:—~~~~
3308:with:
2902:videos
2898:carbon
2644:means
2546:means
1952:JDiala
1900:=: -->
1799:(talk)
1794:). —
1534:Change
1333:JDiala
1203:Simple
352:WP:DUE
136:, the
4202:Limit
3807:over
3762:IGNOU
3453:x - x
3436:Axiom
3209:|ans=
3195:This
2777:with
2277:Nice
1480:limit
1293:Jerzy
1233:. We
769:, so
510:, if
442:to ||
313:WP:OR
149:limit
16:<
4346:talk
4316:talk
4271:talk
4214:talk
4120:talk
3733:nand
3707:nand
3656:talk
3615:talk
3536:talk
3490:talk
3418:talk
3403:talk
3172:talk
3153:talk
3122:talk
3027:and
3011:talk
2947:the
2888:math
2859:talk
2829:talk
2602:and
2496:not
2474:talk
2422:talk
2283:talk
2268:talk
2264:Ozob
2249:talk
1971:talk
1956:talk
1928:talk
1849:talk
1845:Ozob
1829:talk
1782:talk
1778:Ozob
1626:talk
1564:talk
1544:talk
1518:talk
1514:Ozob
1503:talk
1463:talk
1442:talk
1438:Ozob
1398:talk
1370:talk
1337:talk
1217:this
1161:talk
1157:Ozob
1145:talk
1095:talk
1079:talk
1075:Ozob
1038:talk
588:and
473:talk
469:Ozob
446:||f(
410:) →
369:end.
360:talk
302:talk
298:Ozob
163:talk
132:, a
116:talk
112:Ozob
95:talk
4273:):
4177:or
3791:in
3746:- "
3329:lim
3239:lim
3207:or
3199:to
3089:. —
2718:If
2502:all
2488:to
2349:RfC
2319:to
1962:".
1776:.
1608:or
1602:or
1235:may
1073:?
1030:.)
467:.
450:/||
264:in
245:at
228:of
4348:)
4318:)
4308:}}
4302:{{
4298:}}
4292:{{
4216:)
4185:Δ
4122:)
4079:−
4000:−
3874:−
3782:AI
3726:ev
3700:ev
3658:)
3650:--
3621:)
3617:•
3538:)
3492:)
3481:.
3469:.
3420:)
3405:)
3365:−
3336:→
3275:−
3246:→
3213:no
3174:)
3155:)
3147:?
3124:)
3077:,
3013:)
2993:3.
2904:)
2900:•
2890:}}
2886:{{
2861:)
2831:)
2664:∘
2655:↦
2613:↦
2575:⋅
2557:↦
2515:↦
2480:)
2476:•
2447:∘
2424:)
2362:.
2357:}}
2353:{{
2285:)
2270:)
2251:)
2217:∂
2209:∂
2192:β
2178:α
2174:∫
2156:α
2135:α
2120:−
2106:β
2081:β
2026:β
2012:α
2008:∫
1973:)
1958:)
1934:)
1930:•
1851:)
1831:)
1792:+1
1784:)
1646:∫
1628:)
1566:)
1546:)
1520:)
1505:)
1465:)
1444:)
1400:)
1372:)
1339:)
1291:--
1231:}}
1225:{{
1163:)
1147:)
1097:)
1081:)
1040:)
742:∈
723:−
611:−
521:∈
475:)
420:/
406:,
385:=
362:)
341:=
304:)
274:/
254:−
216:,
165:)
118:)
97:)
4344:(
4314:(
4269:(
4254:)
4251:t
4248:(
4212:(
4204:.
4188:x
4175:h
4154:)
4151:t
4148:(
4118:(
4100:.
4095:h
4091:)
4088:a
4085:(
4082:f
4076:)
4073:h
4070:+
4067:a
4064:(
4061:f
4044:h
4029:0
4026:=
4023:x
3997:=
3994:x
3974:0
3971:=
3968:x
3948:0
3945:=
3942:x
3930:h
3913:h
3895:.
3890:h
3886:)
3883:a
3880:(
3877:f
3871:)
3868:h
3865:+
3862:a
3859:(
3856:f
3836:h
3730:A
3724:D
3721:—
3704:A
3698:D
3684:s
3681:r
3678:i
3675:f
3672:b
3669:D
3654:(
3613:(
3534:(
3518:s
3515:r
3512:i
3509:f
3506:b
3503:D
3488:(
3479:1
3475:1
3467:1
3463:1
3459:
3455:1
3450:/
3447:)
3445:1
3440:
3416:(
3401:(
3381:h
3377:)
3374:a
3371:(
3368:f
3362:)
3359:h
3356:+
3353:a
3350:(
3347:f
3339:0
3333:h
3325:=
3322:L
3291:h
3287:)
3284:a
3281:(
3278:f
3272:)
3269:h
3266:+
3263:a
3260:(
3257:f
3249:0
3243:h
3235:=
3232:L
3170:(
3151:(
3120:(
3055:n
3050:R
3009:(
2990:=
2987:n
2965:n
2960:R
2945:n
2929:—
2896:(
2878::
2874:@
2857:(
2827:(
2812:.
2808:″
2805:g
2801:=
2796:)
2793:2
2790:(
2786:g
2763:2
2759:g
2735:2
2731:g
2720:g
2706:.
2703:)
2700:)
2697:x
2694:(
2691:g
2688:(
2685:g
2682:=
2679:)
2676:x
2673:(
2670:)
2667:g
2661:g
2658:(
2652:x
2632:)
2629:x
2626:(
2621:2
2617:g
2610:x
2590:,
2587:)
2584:x
2581:(
2578:g
2572:.
2569:)
2566:x
2563:(
2560:g
2554:x
2532:2
2528:)
2524:x
2521:(
2518:g
2512:x
2472:(
2450:g
2444:g
2420:(
2394:)
2390:(
2377:.
2370:.
2281:(
2266:(
2247:(
2229:t
2226:d
2220:x
2212:f
2201:)
2198:x
2195:(
2187:)
2184:x
2181:(
2170:+
2164:x
2161:d
2153:d
2147:)
2144:)
2141:x
2138:(
2132:,
2129:x
2126:(
2123:f
2114:x
2111:d
2103:d
2096:)
2091:)
2088:x
2085:(
2078:,
2075:x
2071:(
2067:f
2064:=
2061:t
2058:d
2055:)
2052:t
2049:,
2046:x
2043:(
2040:f
2035:)
2032:x
2029:(
2021:)
2018:x
2015:(
2001:x
1998:d
1994:d
1969:(
1954:(
1926:(
1906:r
1847:(
1827:(
1780:(
1762:3
1758:x
1754:=
1751:)
1748:x
1745:(
1742:f
1718:,
1713:2
1709:)
1705:x
1702:(
1699:f
1694:2
1691:1
1686:=
1683:x
1680:d
1675:)
1672:x
1669:(
1665:′
1662:f
1658:)
1655:x
1652:(
1649:f
1624:(
1562:(
1542:(
1516:(
1501:(
1461:(
1440:(
1396:(
1368:(
1335:(
1313:œ
1297:t
1295:•
1190:œ
1159:(
1143:(
1093:(
1077:(
1059:1
1055:C
1036:(
1016:1
1012:C
989:2
984:a
980:m
959:a
937:2
932:a
928:m
905:a
901:m
874:2
869:a
865:m
860:/
854:a
850:m
827:1
823:C
800:a
796:m
792:=
787:2
782:a
778:m
755:2
750:a
746:m
739:f
734:2
730:/
726:1
718:|
713:f
709:|
703:2
699:/
695:1
690:|
685:f
681:|
677:=
674:f
652:a
648:m
627:f
622:2
618:/
614:1
606:|
601:f
597:|
574:2
570:/
566:1
561:|
556:f
552:|
529:a
525:m
518:f
496:0
492:C
471:(
464:a
460:m
456:f
452:x
448:x
444:x
440:x
436:f
432:a
426:a
422:m
417:a
413:m
408:R
404:S
402:(
400:C
396:R
391:a
387:m
382:a
378:m
358:(
347:a
343:m
338:a
334:m
330:a
328:(
326:f
321:a
317:m
300:(
294:S
290:R
286:R
280:a
276:m
271:a
267:m
262:)
260:a
258:(
256:f
252:f
247:a
243:f
238:a
234:m
230:R
226:a
222:R
218:R
214:R
212:(
210:C
161:(
114:(
93:(
50:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.