Talk:Inverse function rule

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

Discussion of Rewrite

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

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?

- The first proof included the following

which combines to:

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

{\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.

"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.

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

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.

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

Good idea - lets shift the page then shall we?

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

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.

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

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

hawthorn