Talk:Inverse function rule

Old arguments deleted. Look `em up in the history pages.

Before we get into an edit war - if anyone really doesn't like the rewrite I would hope we could discuss it first.

The basic question that prompted this is what is the page about. Supposedly it is about inverse functions and differentiation. Not inverse functions (all my technical stuff and examples). Not differentiation. But the two concepts together, which can only mean the reciprocal relation between the derivatives.

The idea is to get to the point as fast as possible, point people who want to know more about inverse functions and calculus at the right places, and then gives some examples not just of inverse functions but of the whole reciprocal thing.

Only the one really simple proof is needed in my opinion. And lets make full use of the chain rule - people who want details can look there.

The integral formula stuff for the inverse function had errors in it. I've fixed it up so that it is now true. Functions can be differentiable but non-invertible - even locally!

Best Wishes 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. CSMR 07:48, 4 May 2006 (UTC)[reply]

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I think that the new version is much worse. It is not pedagogoical and not helpfull at all for someone who just wants to learn what an inverse function is. Please revert. User:Kevin_baas

Wouldn't someone who just wants to know what an inverse function is be better off looking at page inverse function? (suitably upgraded if neccessary)

Even accepting your point of view, I wouldn't recommend a simple revert. The old page had

• Examples of inverse functions, but no examples talking about their derivatives
• two proofs, the first of which was in my opinion flawed.
• some fallacies about the relationship between differentiability and invertibility.

Forthese reasons I would prefer to discuss which aspects of the old page you want preserved and talk about reintroducing just those elements.

hawthorn

In respone to your points:

• Examples of the derivatives of inverse functions were under development(by pizza), and there was a perfect spot for them at the bottom. Thank you for making examples. These are perfect for this application.
• the advantage of two "proofs" is obvious. "your opinion" is quite vague, and to use the word "flawed" is inappropriate, being that they are mathematically correct (hence you said "my opinion"). Furthermore, the first "proof" is geometrically clear and intuitive. How is that a flaw?

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• • The first proof included the following

which combines to:

${\displaystyle f'(x)={\frac {df(x)}{dx}}={\frac {dy}{dx}}={\frac {dy}{df^{-1}(y)}}={\frac {1}{{f^{-1}}'(y)}}}$

The last step cannot be justified without making use of the result we are trying to prove, namely

${\displaystyle {\frac {dx}{dy}}={\frac {1}{\frac {dy}{dx}}}}$

Note that this isn't just a trivial nitpick on my part. It is a fatal flaw. The proof actually uses circular logic. The Leibnitz notation may look like a fraction, but it isn't actually a fraction. You can't assume it has the same inverse = reciprocal behavior without proof.

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• "some fallacies", is quite a politically tainted expression. To say that it is fallacious is misleading. The difference between continuous and differentiable, and differentiable, is slight, especially given the fact that were a function is not continuous, it is not differentiable.(however, the converse does not always hold) In any case, this is quite a small point, which could be easily 'tweaked' by a very very minor correction.

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• • Politics has nothing to do with it. This is maths. Consider the function

${\displaystyle y=x^{2}\sin \left({\frac {1}{x}}\right)+x}$

This function is differentiable everywhere. Its derivative however has an essential discontintuity at zero. Note that the derivative at zero is 1, but the function has no local inverse in a neighbourhood of zero. You might argue that the problem only occurs at one point. But by cunningly adding examples like this together, I can build examples where the same bad behaviour happens at many points - at countably many points even - maybe even at every rational point, although that might take more work. Invertible and differentiable are quite distinct concepts, and the relationship between them is a lot more subtle than you seem to think.

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On the point of the page title: the title should then be "differentation of inverse functions", rather than the ambigious title.

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• Good idea - lets shift the page then shall we?

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But let's stop a moment here, could the two benefit from being combined into one page, or would they be better off separate?

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• I'd say better off separate. I don't really see all that much synergy between the two topics. The only link really is the law which is the focus of the amended page.

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I would argue that the concepts are simple and belong together, if they only want to learn about inverse functions, they can stop at half the page. If they only want to know about differentation thereof, they can skip the first half, and they have a reference with a smooth transition, so that the approach is contextualized. Similarily, the approach from inverse->diff is straightforward. Besides, one should generally introduce a topic before they discuss it.

User:Kevin_baas 2003.06.28

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• I suspect you think there is a closer relationship between the two than actually exists.

hawthorn

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Re:

${\displaystyle {\frac {dx}{dy}}={\frac {1}{\frac {dy}{dx}}}}$

implicit differentation? In any case, I can see this spatially, I could see this spatially in high school. It's trivial.

Re: "I suspect you think there is a closer relationship between the two than actually exists." Probably not. I think that the relationship is just very straight-forward to me. I figured it out on my own in high school.

Also: I don't think this is the proper page to show that trivial result. It's really very trivial, and it really is dividing. dx isn't just a symbol, it's a function of x.

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. -kb

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

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. -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? -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.

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.)-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?

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

• 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. -kb

What do you think is missing. 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

${\displaystyle {\frac {dx}{dy}}\,.\,{\frac {dy}{dx}}=1}$

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. -kb

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. --69.91.95.139 (talk) 03:13, 6 February 2008 (UTC)[reply]

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.--Gustav Ulsh Iler (talk) 03:21, 24 October 2009 (UTC)[reply]

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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.

-Tomhallward(talk)

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.