Talk:Calculus with polynomials

AxelBoldt edited the proof for this page. However, while his proof is more standard, it only applies for r a positive integer, whereas my proof can be extended to the whole complex plane. Essentially write x^n as exp(nln(x)) and differentiate, since the derivatives of e^x and ln(x) respectively are trivial to prove in the complex plane.

But the title of the article speaks of polynomials. If r is not a positive integer, then x^r is not a polynomial funtion. 193.48.101.101 15:03 19 Jun 2003 (UTC)

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The derivative of e^x in the complex plane is trivial if you define e^x as a power series, and then differentiate the power series term by term. In order to do the latter, you need to know the derivative of x^k for positive integers k. AxelBoldt 16:24 Dec 10, 2002 (UTC)

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Is it too much to expect User:Hawthorn to explain his edit? Pizza Puzzle

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I explained my edit comprehensively on your talk page. Is it too much to expect you to explain why you have decided to reinsert a whole bunch of trash since then. I'll clean up the page one more time, but I have not got the time or patience to come back and do it over and over again. My regular job (as a mathematician!) takes up too much of my time. Please THINK before you reinsert rubbish on this page. Don't insert irrelevant stuff. Don't insert incorrect stuff. Don't put in strange headings. Stick the topic (polynomials). And make at least some effort to be clear and concise and not waffle. Sheesh!

• Deleted rubbish

(If f(x) = x^r, for some real number r; f '(x) = r x^r-1)

It is unneccessary - an example. It is very bad mathematical writing to clutter up a definition with examples. Put them after the definition. Furthermore "for some real number r" is just rubbish. In the definition r ranges from 0 to n so you are contradicting the definition within itself - bad bad idea. Furthermore there are problems with talking about a^r for a general r which you have completely ignored. Think about <math>(-1)^\frac{1}{2}<math> for example. Tell me what <math>(-1)^\pi<math> means? Unless you want to talk about this stuff - and you shouldn't need to on a page about polynomials - then stay away from anything other than positive integer exponents.

• Deleted more rubbish

(except when r = -1)

Since r ranges from 0 to n in the definition, it can never be equal to -1. So the comment is completely unneccessary.

• The heading ```differentiation is unneccessary. You are giving examples of use of the formulae. It doesn't need a separate heading. It should follow straight on from the formulae.
• Only a small number of examples need be given. Not endlessly many. And if you are going to give examples, then you need examples of integration as well as differentiation.
• Deleted more strange stuff

These results can be verified with an understanding of Newton's difference quotient and the binomial theorem. One can also derive the General Power Rule via the Chain Rule. A more complex definition of the GPR, for some real number r and some differentiable function f(x), is: f '(x) = r[f(x)]^{r - 1}(f '(x)) = rf(x)^{r - 1}f '(x). For example, if f(x) was 3x¹; then f '(x) = 1 · f(x)⁰ · 3 = 3.

The first sentence is unneccessary since a proof is given later. Your more complex GPR is just a direct consequence of the chain rule. In any case you are back to your `for any real r' habits again, completely ignoring the problems that arise for anything other than positive integer r. This stuff shouldn't be on this page. This page is ostensibly about differentiation of polynomials. These are not polynomials. They don't need to be here.

• Deleted Informal Proof section.

==Informal Proof==

Consider the derivative of x^r; where r is a positive integer, greater than 1. This derivative (as h approaches 0) equals: [(x + h)^r - x^r] / h (the difference quotient); which (as h approaches 0) equals: x^r + rx^{r - 1}(h) + {[r(r - 1)x^{r - 2}] / 2}(h)² + ... + (h)^r - x^r} / h (a binomial expansion); which equals (as h approaches 0): x^r + rx^{r - 1} - x^r; which equals rx^{r - 1}.

My question is, why is this even here. There is a good proof below. That proof is pretty much the same as this one. Why repeat it? Better to give one nice simple proof, not a whole bunch. This one is not as clear as the proper one which follows. You talk about negative r again, which is unneccessary on a page about polynomials. The page is about polynomials, so why are you even mentioning other than positive integer exponents.

• Fixed heading on Proof. The section gives a proof of the derivative formula. It isn't about linearity. I don't understand why you want to put this stupid heading here.

OK. If you want to actually improve the page ...

• It could do with a section of links to related topics. In particular it needs more links to the other calculus pages.
• The C(n,k) notation in the proof should be cleared up, and replaced with more standard <math>{}^nC_k<math> notation.
• A proof for the integration result could be added.
• If you want to talk more about negative exponents and so on, put them in the generalisations section at the end. But keep this very short and brief, and try to give mostly links to other math pages.

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• The heading ```differentiation is unneccessary. You are giving examples of use of the formulae. It doesn't need a separate heading. It should follow straight on from the formulae.
  • By using a heading, the reader can easily skip to a portion of the text which they are looking for.
• Only a small number of examples need be given. Not endlessly many. And if you are going to give examples, then you need examples of integration as well as differentiation.
  • While the number of examples should be limited, there is no reason that one should be required to write examples for both integration and differentiation, this is a work in progress.
• These results can be verified with an understanding of Newton's difference quotient and the binomial theorem. is unneccessary since a proof is given later.
  • It is not unnecessary. The later proof does not link to Newton, his quotient, or the binomial theorem.
• Your more complex GPR is just a direct consequence of the chain rule. This stuff shouldn't be on this page.
  • Perhaps so, but it would be more productive if you would move the text to chain rule, instead of deleting it.
• This page is ostensibly about differentiation of polynomials. These are not polynomials. They don't need to be here.
  • A good point, I came to this page from a link which was not ostensibly about polynomials. Perhaps a new page should be created.
• Deleted Informal Proof section. My question is, why is this even here. There is a good proof below. That proof is pretty much the same as this one.
  • The proofs differ notably in appearance and terminology, a reader may not understand the terms for one proof, but might understand those for the other.

Pizza Puzzle