Talk:Differentiation rules

completed merge, some assistance required

I merged List of List of differentiation identities here except for this:

For higher derivatives the chain rule is given by Faà di Bruno's formula (below is the combinatoric form):

${d^{n} \over dx^{n}}f(g(x))=\sum _{\pi \in \Pi }f^{(\left|\pi \right|)}(g(x))\cdot \prod _{B\in \pi }g^{(\left|B\right|)}(x)$

This belongs in the Nth derivatives section, but someone who knows more about math needs to do this. D O N D E groovily Talk to me 17:39, 6 February 2011 (UTC)

Possible error ?

shouldn't the n'th derivative of the power x^N be proportional to x^(N-n) instead of the product index r ? —Preceding unsigned comment added by 132.187.40.95 (talk) 11:20, 11 April 2011 (UTC)

Thanks very much for finding that ! ... Maschen (talk) 12:20, 8 June 2011 (UTC)

The polynomial or elementary power rule

Right now this section states: "If $f(x)=x^{n}$, for some natural number n (including zero) then

$f'(x)=nx^{n-1}.\,$"

But this isn't only true for $n\in \mathbb {N}$; it's true for any $n\in \mathbb {C}$ (and of course $\mathbb {R}$, $\mathbb {N}$, $\mathbb {Q}$, $\mathbb {Z}$ etc.). — Preceding unsigned comment added by Philmac88 (talk • contribs) 20:54, 16 August 2011 (UTC)

Eliminate prime notation - use Leibnitz' powerful d

This is not just my personal comment. Prime notation for derivatives is:

$f'=\mathrm {effing\,stupid} \,$

because:

• its easy to loose this when writing for a dot, splodge, smudge etc. and may lead to errors,
• it is unclear what to differentiate with respect to, context needs to provide explanation instead of stating the variable. With d notation only the variables and their symbols need to be stated - then its absolutely clear what dy, dy/dx is, the infinitesimal amount simply has a d in front, using subscripts just clutters notation, which should be used for index variables or naming variables,
• more intuitive: this is not actually what d is, but is a correct interpretation: d can be considered as a very very very (...) small number tending asymptotically to zero:

$\mathrm {d} x=\lim _{\delta x\rightarrow 0}\delta x\,$

"Multiplying" by another d gives an even smaller quantity: :$\mathrm {d} (\mathrm {d} x)=\mathrm {dd} x=\mathrm {d} ^{2}x\,$

"Multiplying" by d any number of times gives an even smaller still quantity: $\mathrm {d} (\cdots (\mathrm {d} (\mathrm {d} x)))=\mathrm {d\cdots dd} x=\mathrm {d} ^{n}x\,$

We can also multiply the full dx with itself as many times as needed, raising it to a power:

$(\mathrm {d} x)(\mathrm {d} x)\cdots (\mathrm {d} x)=\mathrm {d} x\mathrm {d} x\cdots \mathrm {d} x=(\mathrm {d} x)^{n}=\mathrm {d} x^{n}\,$

this explains the notation:

${\frac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}={\frac {\mathrm {d} }{\mathrm {d} x}}\left(\cdots {\frac {\mathrm {d} }{\mathrm {d} x}}\left({\frac {\mathrm {d} y}{\mathrm {d} x}}\right)\right)={\frac {\mathrm {d} \cdots \mathrm {d} \mathrm {d} y}{\mathrm {d} x\cdots \mathrm {d} x\mathrm {d} x}}={\frac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}$

• the notion of an infinitesimal quantity is too powerful to ignore. The ratio of two infinitesimal quantities (say) dy/dx is either that or the derivative of y w.r.t. x. The manipulation of multiplying/dividing by infinitesimal quantities can be just that and allow all derivatives to be derived. To derive the ratio of infinitesimal amounts of variables y as a function of x:

$\mathrm {d} x=\lim _{\delta x\to 0}\delta x\,$

left-hand dy

$\mathrm {d} y=\lim _{\delta y\to 0}\delta y=\lim _{\delta x\to 0^{+}}[y(x+\delta x)-y(x)]\,$

right-hand dy

$\mathrm {d} y=\lim _{\delta y\to 0}\delta y=\lim _{\delta x\to 0^{-}}[y(x)-y(x-\delta x)]\,$

left and right dy must be equal for continuity, the right-hand ratio of two quantities (also a derivative)

${\frac {\mathrm {d} y}{\mathrm {d} x}}={\frac {\lim _{\delta y\to 0^{+}}\delta y}{\lim _{\delta x\to 0}\delta x}}={\frac {\lim _{\delta x\to 0^{+}}[y(x+\delta x)-y(x)]}{\lim _{\delta x\to 0}\delta x}}=\lim _{\delta x\to 0^{+}}\left[{\frac {y(x+\delta x)-y(x)}{\delta x}}\right]\,$

left-hand ratio of two quantities (again also a derivative)

${\frac {\mathrm {d} y}{\mathrm {d} x}}={\frac {\lim _{\delta y\to 0^{-}}\delta y}{\lim _{\delta x\to 0}\delta x}}={\frac {\lim _{\delta x\to 0^{-}}[y(x)-y(x-\delta x)]}{\lim _{\delta x\to 0}\delta x}}=\lim _{\delta x\to 0^{-}}\left[{\frac {y(x+\delta x)-y(x)}{\delta x}}\right]\,$

again left and right dy/dx must be equal for continuity. Prime notation hides this: the ratio of two infinitesimal quantities is more general than a derivative. dy and dy x can be treated as algebraic quantities. Ex. the length of a curve L (in 2d Cartesian plane) can be written directly as Pythagoras' theorem:

$\mathrm {d} \ell ^{2}=\mathrm {d} x^{2}+\mathrm {d} y^{2}\,$

$\mathrm {d} \ell ={\sqrt {\mathrm {d} x^{2}+\mathrm {d} y^{2}}}\,$

$\mathrm {d} \ell ={\sqrt {\mathrm {d} x^{2}+{\frac {\mathrm {d} x^{2}}{\mathrm {d} x^{2}}}\mathrm {d} y^{2}}}\,$

$\mathrm {d} \ell =\mathrm {d} x{\sqrt {1+{\frac {\mathrm {d} y^{2}}{\mathrm {d} x^{2}}}}}\,$

$L=\int \mathrm {d} \ell =\int {\sqrt {1+{\frac {\mathrm {d} y^{2}}{\mathrm {d} x^{2}}}}}\mathrm {d} x\,$

• It makes integration much easier, to see which infinitesimal quantities cancel when integrating a derivative. It makes integration by parts and by substitution trillions of times more obvious and easier.

The differential product rule is clear:

$\mathrm {d} (pq)=p\mathrm {d} q+q\mathrm {d} p\,$

${\frac {\mathrm {d} (pq)}{\mathrm {d} t}}={\frac {p\mathrm {d} q+q\mathrm {d} p}{\mathrm {d} t}}={\frac {p\mathrm {d} q}{\mathrm {d} t}}+{\frac {q\mathrm {d} p}{\mathrm {d} t}}\,$

hence integration by parts is clear:

$\int {\frac {\mathrm {d} (pq)}{\mathrm {d} t}}\mathrm {d} t=\int p\mathrm {d} q+\int q\mathrm {d} p=\int {\frac {p\mathrm {d} q}{\mathrm {d} t}}\mathrm {d} t+\int {\frac {q\mathrm {d} p}{\mathrm {d} t}}\mathrm {d} t\,\!$

$\int \mathrm {d} (pq)=\int p\mathrm {d} q+q\mathrm {d} p=\int p\mathrm {d} q+\int q\mathrm {d} p\,$

$pq=\int p\mathrm {d} q+\int q\mathrm {d} p\,$

whatever p or q are, if they can quickly be tucked under the d, integration by parts is really quick.

Integration by sub is obvious:

$y=y(x),x=x(t)\,$

$\mathrm {d} x={\frac {\mathrm {d} x}{\mathrm {d} t}}\mathrm {d} t\,$

$\int y(x)\mathrm {d} x=\int y(x(t)){\frac {\mathrm {d} x(t)}{\mathrm {d} t}}\mathrm {d} t\,$

Just look at the notation alive in action - it’s on the loose, dynamic, manipulating, resembles the familiar elementary algebra, and implies a direct formalism with linear algebra...

As such I'm going to remove all prime notation in this article and use the almighty tools of Leibnitz: the d, and the ∫. Anyone reading this will think I am being sooo pedantic and conscious about notation. But consciousness of powerful notation is a good thing.

Maschen (talk) 18:51, 21 November 2011 (UTC)

Wow - no one has objected after a week. I'll go for it. Also

• i'm going cut down the monotonous repetetive writing in each subsection (e.x. "for functions f and g", "for functions f and g", "for functions f and g" .....) by collecting base formulae (product/quotient/chain/general power rulse etc) together,
• write the definitions of the function in the standard formalism (e.x.):

$f:\mathbf {R} \rightarrow \mathbf {R} \,\!$

to clarify what the function must be for the derivative to exist. E.x. cannot have a function f(x) which vanishes at x₀ (i.e. f(x₀)=0 ) and then calculate the diff of 1/f(x₀), so the mapping would be:

$f:\mathbf {R} -\{x_{0}\}\rightarrow \mathbf {R} \,\!$

for for composition f(g(x)):

$f\circ g:\mathbf {R} \rightarrow \mathbf {R} \,\!$

(I h8 this circle notation - it also looks stupid.....).

I suppose someone will still merrily delete my edits after, even with this much justification for an effort to clarify, not complicate. Please at least see where i'm coming from before doing so.--Maschen (talk) 17:42, 1 December 2011 (UTC)

There's WP:TLDR to consider. And you come off as a little crazy in your post, if you don't mind my saying. Sławomir Biały (talk) 03:17, 2 December 2011 (UTC)

I object, I just didn't notice the comment made a week ago. The current article uses both notations, and it should. Words are generally preferred to formulas for the purposes of learning, but I do not object stenuously to using function notation on occasion. Thenub314 (talk)

In the end I kept both notations. Even still, will you delete the previous two edits I made?--Maschen (talk) 00:25, 2 December 2011 (UTC)

The recent series of edits has many problems. In the first section, "continuous where specified" is irrelevant, as is emphasizing a notation $F^{n}$ that is never even used is irrelevant. Get rid of "Leibniz notation" $df={\text{blah}}$: this is very problematic in this context. It is tempting to go back to the last sane version. What, exactly, is worth keeping here? Sławomir Biały (talk) 02:48, 2 December 2011 (UTC)

I have asked for further input at WP:WPM. Sławomir Biały (talk) 02:52, 2 December 2011 (UTC)

I think the "sources and further reading" added in this edit should be restored. But in all other respects I prefer this version. An encyclopedia article should conform to the commonly used notation, even if it is less than ideal. Regarding the lack of response to User:Maschen's earlier comments: I didn't have this article on my watchlist, and I suspect a lot of other people didn't either. If you're making a major overhaul to an article, it's a good idea to mention it at Wikipedia talk:WikiProject Mathematics. Jowa fan (talk) 05:32, 2 December 2011 (UTC)

The article seems to be intended as a "quick reference" (like an appendix of Shaum's Outline Series), not as a rigorous guide of applicable domains and the like.

• I think the page should remain as focused on essentially the rules one learns first: relating to real functions of one real variable. If need be, the article could be renamed to reflect this focus on single-variable real functions.
• Any expansion to wider areas of applicability such as complex numbers, vectors and matrices (potentially very extensive) should be kept off this page. Any such expansions should be in other similar pages, referenced in a "see also" section.
• A definitions/notations section of more than a sentence or two is a liability. The change so that the first view of the page as full of abstract symbols definitely is a problem.
• The change of function names to uppercase also is unusual; it creates the impression that something specific is meant (e.g. F being the Fourier transform of f). The notation thus being less familiar is not a good idea.
• The domain of applicability can be left to be inferred as it so often is; no more than a simple statement to this effect is needed.
• Keep the power notation to fⁿ and f(x)ⁿ or (f(x))ⁿ (never the operator-like notation for functions fx, fⁿx or fⁿ(x)); that way it is unnecessary to clarify the notation, except to note the exception f⁻¹ as the inverse when used. This fits with the use of the isolated function name as a variable (e.g. f=f(x)) rather than as an operator.
• There may be a case for limiting mixing of notation (df vs. f′). Having repetitions in both notations makes the page less compact and reading more difficult (this is, after all, not a tutorial on the equivalence of the notations); how to reorganize this sensibly is not obvious to me. I imagine that it is common even at undergraduate level to switch freely between these notations according to convenience (compactness vs. algebraic suggestiveness), complicating ths choice.
• It may make sense to limit edits to one type of change at a time; that way other editors can follow what is happening.

Quondum^talk_contr 06:23, 2 December 2011 (UTC)

Fine.

"Regarding the lack of response to User:Maschen's earlier comments: I didn't have this article on my watchlist, and I suspect a lot of other people didn't either."

I truley apologize for not staying up all night staring my eyeballs out at the screen waiting for the off-chance that people will reply, then immediatley responding. I had other things to do (and no it wasn't sleep as you'll assume - writing up solutions to assessements for uni was one of them...). Again - sorry × c².

Furthermore, why is it such a big problem to define nomeclature then proceed through the rest of the article? I've done this a number of times and everyone thinks its unclear. I don't mind that, but why do people thinks it unclear and useless?. How are we supposed to know what is what? Are we really supposed to keep explaining symbols throughout? I know - its just me.

It doesn't matter - enjoy your prime notation... No revert to d will ever be made. No I couldn't care less if all you higher-uppers think fucktards like me are crazy. Irrelavent.--Maschen (talk) 08:22, 2 December 2011 (UTC)

If you're responding to my points, perhaps you didn't read too well. I was saying there was excess clarification introduced. And I didn't not oppose your original point about notation. You unfortunately included unrelated contentious edits at the same time. Quondum^talk_contr 11:24, 2 December 2011 (UTC)

Now I apologize for not being clearer. I was talking to all of you - some of the dominant editors of maths articles (hence the term higher-uppers). Also I forgot to ask - why does Sławomir Biały think the d notation is so "problematic" in this context?

""Get rid of "Leibniz notation" $df={\text{blah}}$: this is very problematic in this context."<---There was no mention as to why its problematic, the sentence simply ended there...why? (here's you're chance to look good and make me seem more stupid than ever)--Maschen (talk) 18:49, 2 December 2011 (UTC)

I'll be presumptious and answer that from my perspective (Sławomir Biały may have meant something different). To use a notation in which d²f (or higher derivatives) appears on its own is problematic, as this notation for higher derivatives does not make algebraic sense (according to Roger Penrose), even though it does at the level of first derivatives. Given this, it may make sense to stick with the notation where it is presented as ratios or operators throughout, e.g. d
dxsin x = cos x instead of d sin x = cos x dx, and df
dx = df
dg dg
dx instead of df = df
dgdg – this is in keeping with Leibnitz's notation being suggestive only. We should in any event not be going out on a limb with the notation if standard texts don't do this. You may be interested in differential forms, where dx is not an infinitesimal, and behaves algebraically as for Leibnitz (but you can't divide by it, and d²=0). Quondum^talk_contr 20:43, 2 December 2011 (UTC)

Very well. Now is probably the best time to close this section. --Maschen (talk) 08:37, 3 December 2011 (UTC)

nth derivatives

In accordance with WP:MOSMATH and the already italic d in this article, I suppose the nth diff section should have d italicised. (b.t.w. I wrote that table - others edited after without actually deleting it). --Maschen (talk) 08:37, 3 December 2011 (UTC)

Furthermore:

• F and G (capital) need to change to f and g (lower) for consistency with the previous setion. (Honestly - i've always seen and used capital and lower case for functions interchangably - capitalized letters of a function don't always mean a Fourier transform of the lower case function, though it is a convention, sometimes for a FT tildes or hats are used).
• The title can be made better: Derivatives to nth order.
• this sentence is a bit funny - I don't remember writing it this way all that time ago (though probably my mistake anyway):

"...real number constants are A, B, N, r, real integers are n and j, ...",

it should be:

"...real number constants are A, B, N, positive integers (used for indicies) are n, m, r, and j, ...".

It might be best to decompose into bullet points instead of evrything all on one line. This much can be done.--Maschen (talk) 09:19, 3 December 2011 (UTC)

Sources and further reading...

There has been a proposal to restore this section, it may as well be launched back into the article. Any article benefits from relavant sources/refs. Also I added a few relevant extra articles under the See also section, not already linked into the context of the article.--Maschen (talk) 09:58, 3 December 2011 (UTC)

Removal of content

I recently removed some of the content of this article with this edit. It was recently brought to my attention that this was merged here from list of differentiation identities. I feel that this content is particularly unsuited to this article, since they aren't considered to be "differentiation rules" in the same sense that the other rules listed here are. It has been suggested that the original list of differentiation identities article should be restored so that this content has a place somewhere. Opinions?

I personally don't think it should be, because it seems pretty indiscriminate to me. Not only that, but it seems to be sourced entirely to someone's empirical observations from a CAS (which makes it WP:OR). Now, I have no doubt that a reference could be found for some of these, but so can references for millions of trivial differentiation identities. We are not a repository for that kind of indiscriminate information. Sławomir Biały (talk) 11:48, 30 December 2011 (UTC)

I think arithmetic calculations made by CAS or a calculator are not an original research, making such calculations is permitted by the rules. But this is anyway related to the formulas for higher-order derivatives for which we can have another list (for example, List of higher-oder derivatives) or include it in a related article.

But you also removed the rules for differentiation of composite and indirect functions which I think should be restored here (and I think we also could add formulas for differentiation of multivariable functions, differentiation of a function in certain direction etc).--Anuclanus (talk) 20:16, 5 January 2012 (UTC)

The idea is not to have a list of every conceivable derivative that could arise, but rather to cover the rules for differentiation as this is commonly understood in textbooks on the subject. I agree that perhaps a mention of the chain rule in several variables might be a good idea, but I wouldn't emphasize too many special cases of it. Sławomir Biały (talk) 23:00, 5 January 2012 (UTC)

The Referances section has been completley emptied... I'll try to find sources for the core rules to fill in at least a couple of cited referances, in addition than those in the further reading section. --Maschen (talk) 12:43, 6 January 2012 (UTC)

PS - good job to everyone who re-wrote the article, since I effectivley screwed it up. Its really clear from months ago. I don't favour or care about the annihilation of my own creation of the N diffs table, of course...--Maschen (talk) 12:54, 6 January 2012 (UTC)

Radians and degrees

Why doesn't the article mention that trig functions must be in radians not degrees for this to work!

Hardly a minor point!

Rosa Lichtenstein (talk) 00:14, 16 February 2012 (UTC)

error in 'Derivatives of exponential and logarithmic functions'?

Derivatives of exponential and logarithmic functions,

(ln x)´ = 1/x , x =/= 0.

I suppose it's defined for x < 0 for complex numbers? Since there's no notation of x < 0 not allowed in R.

The second one, only denoted as (ln |x|)´ = 1/x

but no notion of x =/= 0 is given — Preceding unsigned comment added by 83.250.91.84 (talk) 18:49, 28 May 2012 (UTC)

Antoher Error in 'Derivatives of exponential and logarithmic functions'?

I believe that there's another error in the derivative of ln |x|. The current version of this page says:

${\frac {d}{dx}}\left(\ln |x|\right)={\frac {|x|}{x^{2}}}$

However, I believe that the correct answer should be:

${\frac {d}{dx}}\left(\ln |x|\right)={\frac {x}{|x|^{2}}}$

However, I'm not an expert in maths so was wondering if someone could verify this:

According to this page:

${\frac {d}{dx}}\left(|x|\right)={\frac {x}{|x|}}$

|x|´ = x / |x| for x != 0

Meanwhile, according to This page:

${\frac {d}{dx}}\left(\ln u\right)={\frac {1}{u}}{\frac {du}{dx}}$

Putting these two together, we set u to be |x|, then:

${\begin{aligned}{\frac {d}{dx}}\left(\ln |x|\right)\\={\frac {d}{dx}}\left(\ln u\right)\\={\frac {1}{u}}{\frac {du}{dx}}\\={\frac {1}{|x|}}{\frac {d}{dx}}\left(|x|\right)\\={\frac {1}{|x|}}{\frac {x}{|x|}}\\={\frac {x}{{|x|}^{2}}}\end{aligned}}$

--Liron (talk) 00:43, 19 October 2013 (UTC)

possible error in hyperbolic functions

isn't sinh(x) = (e^x - e^-x)/2, and if so, then the derivative of cosh(x) should equal positive sinh(x) not the negative. 129.138.32.84 (talk) 13:52, 30 September 2015 (UTC)

Languages

The languages menu for the differentiation is really messed up because in some languages it's called derivate

Please take a close look at the language menu of these pages

https://es.wikipedia.org/wiki/Anexo:Derivadas

https://ro.wikipedia.org/wiki/Tabel_de_derivate

https://wiki.riteme.site/wiki/Differentiation_rules

https://fr.wikipedia.org/wiki/D%C3%A9riv%C3%A9es_usuelles

Some pages link to almost no other languages at all and the content even if should be the same is totally different most likely cause some authors can't observe the article in other languages in order to inspire from much better written articles. — Preceding unsigned comment added by 188.25.181.147 (talk) 22:30, 22 November 2015 (UTC)

About the inverse

Why there is no inverse rules main important rules are onlt they Bapi paul07 (talk) 17:57, 9 February 2018 (UTC)

Do you mean this inverse rule: Differentiation rules#The inverse function rule? –Deacon Vorbis (carbon • videos) 18:08, 9 February 2018 (UTC)

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