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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]]
* 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.
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: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]]
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