Talk:Calculus/Differentiation/Differentiation Defined

2003
Why use square brackets for the differential operator? IMHO, round brackets are just fine... Dysprosia 09:17, 16 Aug 2003 (UTC)


 * My calc book uses square brackets when differentiating an expression, like $${d \over dx} [x^2]$$. Some is style, some is real theory.  Ex.: what is $$\lim_{\Delta x \to 0} {\Delta y \over \Delta x}$$?  Why it's $${dy \over dx}$$ of course.  It would look wierd to have $${\Delta \over \Delta x} y$$; delta what, right?  So I guess that's the reason.  However, that's also why I think that the equation for the derivative should be $$\lim_{\Delta x \to 0} {f(x + \Delta x) - f(x) \over \Delta x}$$ because it correctly shows what dy/dx really means.Boothinator 02:17, 19 Mar 2004 (UTC)

question
How do the last two steps listed for f'(sin(x)) work? Ugen64 01:09, 18 Dec 2003 (UTC)


 * You mean D sin(x)? The second summand should be clear (from sin(x)/x -> 1 as x->0), and the second is a result from that limit, though I should point that out in the article (and leave it to the Trig page ;). Dysprosia 05:20, 18 Dec 2003 (UTC)


 * Is it worth it to prove the product rule, division rule, or chain rule here? Do prove the mean value theorem in this book?

--Cronian 07:43, 26 Mar 2004 (UTC)


 * OK, am I the only one who can't see the apostrophe in f'(x)? If not, should I put a space after the f, or is there a better way to make it more readable? --Xhad

If you itallicize the 'f' and the 'prime' you can read it a bit more clearly: f'(x). The whole bit itallicized: f'(x). Maybe that's better? ChaoticLlama  14:59, 13 April 2007 (UTC)

question
In the differentiation rules (d/dx)c=0 and (d/dx)mx=m, what exactly does "d" stand for? does it stand for change in x and y??? could somebody please explain these rules more clearly? --Elpenmaster
 * I think of the letter d as meaning "a tiny difference". ..... (d/dx)c means a tiny difference of a constant (obviously there is none, or else it would not be a constant!) divided by a tiny difference in x is zero. ..... And (d/dx)mx means (a tiny difference of the product m times x) divided by (a tiny difference in x) is m, where m, by the way, is the trigonometric tangent of the angle of that line (y = mx + any constant) against the horizontal. QUITTNER 142.150.49.171 18:42, 17 Jun 2005 (UTC)


 * First: The power rule is used in an example (about the addition rule) before it is introduced. Would people new to calculus be confused by the sudden transformation and miss the point of the example as a whole?


 * Second: Regarding proofs, in my opinion, lengthy proofs (read, "90% of proofs") are not overly important to the average reader. I believe that important proofs (chain rule, product rule, etc.) could all be placed on a seperate page in order to keep the text from becoming cluttered. If the goal is to reach people who don't already understand calculus, too much notation all at once becomes confusing and distracting.

--User:Noah 16:14, 30 Jul 2004 (UTC)

power rule
I noticed that the power rule section said $$\frac{d}{dx}\left[x^n\right]=nx^{n-1}, n\ne0$$. If you work it out when $$n=0$$, you get $$\frac{d}{dx}\left[x^0\right]=0x^{0-1}=0x^{-1}=0$$. Since $$x^{0}=1$$, you get $$\frac{d}{dx}\left[1\right]=0$$, which is true. Unless I made a mistake, the $$n\ne0$$ is not needed. Should it be taken out of the equation? DanielLC 00:28, 1 August 2005 (UTC)

While often $$x^{0}=1$$, there is some confusion over the special case of x=0 (0^0).

The $$n\ne0$$ keeps us far, far away from that one point of confusion.

--DavidCary 20:13, 31 October 2005 (UTC)

bold new words
Should new words like "derivative" and "differentiable" (Section 2.1) be emphasized (bold/italic) for clarity? --anonymous

I think the first thing that should be included on this page is the definition of the derivative of a function by First Principles, ie. limit of the gradient of secants, and then possibly explain some of the derivatives pulled, apparently, out of the blue, in terms of that.

Second, for those asking what the d in (d/dx)c is all about - d/dx is a "differential operator", which means it's something that acts on a function to differentiate it, in this case with respect to x. It follows some of the rules of normal fractions, so you have (d/dx)(f(x)), which means "the derivative (with respect to x) operator applied to the function f of x", and write it df/dx, which means "the derivative of f with respect to x" (ie. pretty much the same thing). Then if f(x) = c, ie. a constant function, then you can write either (d/dx)c or dc/dx, both of which equal 0. 203.26.177.2 18:52, 26 September 2005 (UTC)

High school
I'm a bored high school student currently in Precalc, trying to teach myself derivatives, so hopefully I can contribute to this project. I had to read the old Sales Example several times to figure out exactly what was going on, so I tried to make it more clear, based on what I gathered from the article, and also tried to make it more text-bookesque. Someone please check the changes in that section, because I'm not entirely sure I'm correct. --SimRPGman 03:44, 27 October 2005 (UTC)


 * Well done for deciding to contribute, it is a great way to learn the subject.  That example looks correct to me.  My only concern is the sentence

"Because the functions are defined for all n >= 0 and are only differentiable in n > 0, the only points that can be minima or maxima are those above."


 * Actually it is defined and differentiable for all n. But we are only concerned with positive n (as it makes no sense to produce -5 items).
 * Also here are some idead which might make the example clearer


 * Instead of 8.1n maybe use (8.1)n.  When I first read it I thought it mean 8.(1n)
 * It is implicit by using the integer n that we are dealing with whole numbers. (otherwise we should use a letter like x,y,z).    Since the max is 3.97.... the seller really has to produce 4 units (as that is the nearest whole number) Juliusross 11:34, 27 October 2005 (UTC)


 * The previously quoted sentence was like that when I had it.
 * The example states that all amounts are in thousands, so 3.797 thousand units would be a valid amount at 3,797 units.
 * --SimRPGman 23:09, 27 October 2005 (UTC)


 * My mistake, you are right. Juliusross 23:52, 27 October 2005 (UTC)

Confused
Hey folks, the section on the Maxima and Minima confused the heck out of me. I tried to re-word the first paragraph, as I thought I grasped the point, but the second paragraph eluded me. If someone with more math knowledge than me (I'm trying to learn this stuff) could take a look I'd appreciate it. It almost seems as though it were written by someone with English as a second language, or just someone who knows their math but perhaps isn't as smooth with English. Not that I'm criticising the effort! Thanks all.--Dave
 * Agreed. These two paragraphs need to be expanded (it might take a few pages in an ordinary textbook).  One way to learn is to write.  Do you want to get involved? Juliusross 12:43, 31 October 2005 (UTC)
 * (This is me, the same as above--I just got an account) Well, that sounds like a challenge...sure, I'm game. Let me give it a shot and then get a critique. Dubitable 13:52, 31 October 2005 (UTC)

Also on the subject of Maxima and Minima, I "stole" some info from Wikipedia and formed it into a introduction to the concept. A picture would really be nice. Should we use Wikipedia's picture? --SimRPGman 05:41, 7 November 2005 (UTC)
 * Sure, take any material from wikipedia if you think it is suitable. You should maybe add a line at the bottom of the page to say you have taken material from wikipedia, and put in a link (although I am not 100% sure what is really necessary) Juliusross 02:18, 8 November 2005 (UTC)

Ok, I took a swing at that last paragraph on Maxima and Minima and actually made two paragraphs out of it. Please review and see if it's any clearer and if I'm right (because I'm not entirely certain I am). --SimRPGman 04:27, 21 November 2005 (UTC)

Whoever is editing this book needs to stop adding "Example" as a category - it appears in the table of contents and looks very cumbersome. If each subcategory extended for more than page - THEN it might be good to add example to the table of contents - but as it stands it just looks horrible. I fixed a couple of them. Fresheneesz
 * I agree that this is not optimal. But  we need a consistent way of presenting examples.  Until per page css is possible this is the best I can come up with.  Have a look at the discussion page on the main calculus page and you will see some other comments about this Juliusross 18:40, 5 November 2005 (UTC)
 * You changed my mind. I will not longer use example as a category. Juliusross 02:18, 8 November 2005 (UTC)

I rewrote the Maxima and Minima section (and renamed it, though I'm not sure that it shouldn't be "Stationary Points"), as it was confusing and incorrect in several places. It is possible to classify a stationary point without doubt. My browser crashed and I forgot to log in so I am 87.194.203.205. Sorry. Inductiveload 20:40, 23 October 2007 (UTC)

Better exercises needed
I've already been through the mill on learning derivatives and can practically do them in my sleep (I'm a AP Calculus C student), but I had to read the exercise at the bottom of the page twice to understand what it's asking. To someone who's just learning differentiation it's a horribly ambiguous question to ask. I've added some simpler straightforward differentiation exercises above the existing one, which is now at the bottom of the list (if someone could clean up my LaTEX equation syntax, I'd appreciate it; I've never used it myself). It's better to go through the process itself a few times from a few different approaches before getting that in-depth into the workings of it--asking a student to use the definition of derivative to prove the theories discussed in the article before having them actually do any differentiation independently of running it through the definition of derivative is a bit of a stretch. I also reworded the exercise that was already there to make it a little more clear in what it's asking. --Jesse Hannah 05:38, 24 February 2007 (UTC)

What is Differentiation?
This section is a little bit in depth for someone who is just being introduced to derivatives. While those of us who have already taken calculus would understand it fine, I am a little worried that those who have not taken calculus will be lost as soon as you start introducing syntax. I think this section should just mention that differentiation is a way of solving the tangent problem, and leave it at that. Maybe mentioning the difference quotient would be good. Tapioca 19:23, 5 March 2007 (UTC)

Overall Structure
I've been reading through the differentiation article several times making edits as I go, however I've noticed there is little flow in this article; Even from sentence to sentence. As I finish finalizing the notes I am creating, I will post the structure and teaching method that I believe to be most effective. I would appreciate any comments in modifying this layout, yet do not reply if you merely wish to comment on this post.

ChaoticLlama 17:51, 20 March 2007 (UTC)

Section Removed
I don't see how this hand-waving argument adds to the article. Any comments otherwise? ChaoticLlama 14:53, 13 April 2007 (UTC)

Moving Sections Around
IMO, this article should be focused on describing the theory of differention, and its techniques. Topics such as Maxima & Minima or the Sales Example are clearly applications of differentiation. I'd move Max & Mins to applications, and put the sales example under Mean Value Theorem. Thoughts? ChaoticLlama  15:09, 13 April 2007 (UTC)


 * Well, I agree sales and extrema are application, but the concept of local extrema, and their relation to differentiation, is very important. So if and when they are moved, they should receive their own clearly defined section, preferably at the very top of the application section. cheers. --Cronholm144 07:56, 18 June 2007 (UTC)


 * I agree with both. The extrema section is (now) well written, but it doesn't flow with the rest of this page. More importantly it relies on Calculus/Higher Order Derivatives, which is a later module. --Mrwojo (talk) 01:16, 29 December 2008 (UTC)

addition and subtraction
In the very first line of the section on addition and subtraction rules, the brackets in the expression in the numerator are imbalanced. There are 2 open brackets and 1 close bracket. I would think we need a close bracket just to the left of the first minus sign. $$ \lim_{\Delta x \to 0}\left[\frac{[f(x+\Delta x) \pm g(x + \Delta x) - [f(x) \pm g(x)]}{\Delta x}\right] $$ Johnor (talk) 02:39, 16 October 2008 (UTC)


 * Fixed. --Mrwojo (talk) 01:04, 29 December 2008 (UTC)

Unusual Terminology
Hi, I find the use of the term "Gradient" a bit unusual. I have not often encountered the phrase, and most calculus books I have encountered stick with the term slope. I am fine with keeping the first sentence of section "The Definition of Slope" but I think after that it would be wise to use the term slope. Does anyone mind if I make this change? Thenub314 (talk) 11:08, 21 January 2009 (UTC)

Proof of the Power Rule
I don't have time to implement this now, but if someone else would like to, it would be nice to have a brief discussion of where the power rule for non-integer exponents comes from. There is a proof given at PlanetMath that may be useful. --Greenbreen (discuss • contribs) 17:48, 11 June 2011 (UTC)