Talk:Inverse function rule: Difference between revisions

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Best Wishes [[user:hawthorn|hawthorn]]

Te one "proof" of course assumes differentiability of the inverse, and in showing differentiability of the inverse you will have found the derivative, so is more useful as an aide memoire than a proof. I have made some slight adjustments and added a statement about the differentiability of the inverse which would seem to be required.

[[User:CSMR|CSMR]] 07:48, 4 May 2006 (UTC)

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And: I don't think your little explanation of differentation at the the top is really going to do any good. I think they'll just have to go to the diff. page for that. That is, I think that this page should assume knowledge of differentation as a prerequisite. The top could tell them that they need differentation and provide a link.

[[User:Kevin_baas]] 2003.06.29

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*Implicit differentiation = ? How?

:or the chain rule. The point is, the infinitesimals are not operators, they act just like variables and functions. -[[User:Kevin_baas|kb]]

::In some respects yes. In pther respects no. In all respects - proof required.

[[user:hawthorn|hawthorn]]

*With regard to the spatial visualisation of inverse=reciprocal for derivatives. - sure! That might be a useful addition to the page so long as it is not presented as a proof. Keep in mind though that visualisation can lead us astray as the functions we visualise tend to be 'nice'.

:visualization is the most fundamental tool of mathematics. I don't know what you guys are doing with this stuff if it has nothing to do with spatial reasoning. These aren't just a bunch of symbols that you manipulate and talk about, you know. -[[User:Kevin_baas|kb]]

*dx is not a function of x. It is a differential, originally defined as representing an infinitesimal variation in x. Derivatives as originally defined were simply fractions of these things while integrals were infinite sums. However after Berkeleys attack on the notion of infinitesimal, differentials became regarded as dangerous items when appearing by themselves, and most people preferred to talk about them only when safely confined in a derivative or integral. This situation still holds today.

:by function of x, i do not mean to be formal. i was more to the point in resp. to implicit diff. I hope I don't have to talk in math in order to talk about it, that would be senseless.

:I'm not familiar with Berkeley's attack. I can say, however, that I don't consider him much of a philosopher. Re: Safely confined to a derivative or integral -> well, duh! It's subjective to that context and only has form with respect to it. Whoever was screwing around with infinitesimals like this must have been really clumsy mathematicians. But should we all suffer for this? -[[User:Kevin_baas|kb]]

:: Newton and Leibnitz `screwed around' with this, and I doubt they could be called `clumsy'. Berkeley was a crap philosopher I agree. He had very strange beliefs especially where science was concerned. In particular he believed that science should be concerned purely with observing and cataloguing the world, and that under no circumstances should any attempt be made to explain it. For this reason he hated the success of Newton who was very good at explanations. He also thought that science was 'getting uppity' and trampling on the toes of religion. His article "The Analyst" was nothing short of a full blooded attack on mathematics and science. He found one weak point in the calculus and exploited it for all he was worth. Much as I dislike the motivations and philosophy of the man, the weakness he pointed out was real enough, although in my opinion the retreat to formalism which it caused was an overreaction.

[[user:hawthorn|hawthorn]]

::: Thanks for the info. It is interesting. I wouldn't call them clumsy at all. Btw, it's nice to see that we agree on many things. (Regarding Berk and the reaction.)-[[User:Kevin_baas|kb]]

::: Any mechanism of a scientific theory can be interpreted as a cataloguing of observations (ambiguity aside) so if Newton was "good at explanations" which are opposed to "cataloguings and observings" these explanations could not have been scientific theories but interpretations of scientific theories, so you are saying that Newton was very good at the interpretation of scientific theory, in particular very good showing that scientific theory "explains" rather than describes, and very good therefore at refuting Berkeley's idealism?

[[User:CSMR|CSMR]] 07:48, 4 May 2006 (UTC)

* You might be right about the explanation of differentiation. I felt that something needed to be said - a one sentence brief description of some sort. But it is really hard to write a good one. If you can write a better one - go for it.

:I don't think anything usefull can be said in one sentence, or even a paragraph. They'll just read it, and think that they should at that point be able to understand the rest, and they won't, and they'll just get frustrated. If I'd rewrite it, I'd defer more than explain. If that's fine, I'll go ahead.

:Addition: The page, as it stands, says very little about how inverse functions relate to differentation. I don't know who started this page, or what their exact intentions were, but I don't think that this would satisfy them. -[[User:Kevin_baas|kb]]

:: What do you think is missing. [[user:hawthorn|hawthorn]]

::: A decent-size section which explicitly shows the relationship. Something akin to my "proofs" you were refering to earlier. You asked what from the old should be added back in...

:::** Not the one with the error in it I trust. I think your idea of describing the geometric picture was a better one. In particular some mention of the fact that the graphs inverse functions are reflected in the line y = x , and hence the m₁m₂ = 1 relationship for slopes of lines reflected in the line y = x implies the result pretty strongly.

:::Also, the first two formulas that aren't indented, that's been distracting. I don't know what it is exactly, maybe it's just indenting, maybe putting the discriptions before, maybe I just don't like seeing formulas so small being so prominent.

:::** If you are talking about the description of dy/dx and dx/dy then I agree they dont look good. It is the fault of the way wikipedia typesets maths inline.

:::** I really wanted to put a box around the formula

:::::<math>\frac{dx}{dy}\,.\, \frac{dy}{dx} = 1 </math>

::::which is what the page is all about. However my initial attempts didn't work so I left it. I really would like to put a box around this though.

:::Very minor; I don't care, really: I'm not used to the dot. Is it really better to have it, or to omit it?

:::** feel free to kill it if you want. I used it simply as a spacer because things looked cluttered without it.

::: I think it's clearer without, but that's probably because I'm more acquanted with it's abscence. But then again, that's because it usually is absent. If, by itself, it's affect is purely a conditioned effect, then I would argue that it's omission is more clear, simply because it's one less symbol. -[[User:Kevin_baas|kb]]

== Inverse of y = e^x? ==

Isn't the inverse of y = e^x y = ln(x) and not x = ln(y)? -beasty401

:If, by any chance you are still looking at this page: both are valid. x = ln(y) describes the exact same ''relationship'' as y = e^x, though, whereas y = ln(x) describes the inverse relationship (x and y swapped). When working with inverses, you may swap the x and y at the beginning, or you may swap them at the end. Either way works. Not ever swapping them works too, depending on your goal. --[[Special:Contributions/69.91.95.139|69.91.95.139]] ([[User talk:69.91.95.139|talk]]) 03:13, 6 February 2008 (UTC)

== Completely Empty ==

This is disapointing, almost nothing is here, this is not even the dust of the tip of the iceberg. No theorems of inverse functions are here, nothing is mentioned about the matrix derivatives, its inverse, nonsingularity of the matrix, the one to one property related to the nonsingularity, boundness and openness of the inverse matrix and thus the inverse derivative of the function and the function itself, the continuity of both functions, inverse matrix of the derivative function and the derivative is no where to be found, and the list goes on. Truely disapointed.

I also highly urge that Euler's notation should be inroduced and the concepts be developed using it.--[[User:Gustav Ulsh Iler|Gustav Ulsh Iler]] ([[User talk:Gustav Ulsh Iler|talk]]) 03:21, 24 October 2009 (UTC)

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This is my first time encountering this page. The material is helpful, but lacks citations and seems like a loose end. I propose paring it down and adding it as a subsection to the page on the inverse function theorem. What do folks think about that?

Also, what are some good references for this material? I think that a reference in the spirit of spivak or rudin would be good, in addition to a more elementary source.

-[[user:Tomhallward|Tomhallward]]([[User talk:Tomhallward|talk]])

== Notation is extremely confusing ==

Since there are two variables in the introductory section, there should be at least one notation that refers explicitly to which variables the derivative is with respect to. Since this is an introductory page, writers should not assume readers will pick up on implicit (really, lazy) notation. <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/68.84.199.178|68.84.199.178]] ([[User talk:68.84.199.178#top|talk]]) 05:37, 17 November 2018 (UTC)</small> <!--Autosigned by SineBot-->