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Genesiiis

Differentiation Proofs! - Images Working

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Hi everyone im looking for some help on a couple of differential proof problems. Both I know have a variety of proofs, be it logs or applying rules(product etc), however im looking to prove these more from first principals I and I can not figure out in the first case how to do it, and in the second why something is so, which will make sense when I actually show you my stuff so far. Differentiating e^u where u is a function of x: By other proof is shown to be: ImageHost.orgi ImageHost.orgii So starting from the difference quotient ImageHost.orgiii ImageHost.orgiv ImageHost.orgv ImageHost.orgvi ImageHost.orgvii now applying limits ImageHost.orgviii ImageHost.orgix ImageHost.orgx So my question here is where have I gone wrong. I have looked over this over and again, and I cant see anything wrong with it as such, but there must be something I am not doing??? secondly is an issue I have with a proof of the product rule. This again starts with the difference quotient (as I said I know there are other proofs, but the question regards this one) ImageHost.org i ImageHost.org ii ImageHost.org iii ImageHost.org iv ImageHost.org v ImageHost.org vi Now by this proof if you apply limits the first two terms of vi become a function multiplied by a dirivative, but the thrid term is considered to equal 0 as follows ImageHost.org vii which when the limit is applied to vi you get the product rule! ImageHost.org viii this is what I dont understand, if the limit of [(delta u)/(delta x)]*[(delta v)/(delta x)] considered 0, then why isnt u*[(delta v)/(delta x)] considered 0 also as delta x --> 0, same applies to the second term of vi. Thanks to everyone who looks, I have tried to find answers myself, but I just dont get anywhere (and please dont post if your just going to offer an alternative proof, I know others exist, but I want to know the specifics of these)

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Alrecenk is correct. That is not true unless f(x) is a linear function.

I think you are approaching this problem incorrectly. You should start by first considering the natural logarithm function:

ln(x) = integral(dx/x)

It's obvious just by differentiating both sides that (ln(x))' = 1/x.

The function exp(x) is defined as the inverse of the natural log function. Consider:

y(x) = exp(f(x))

To find the derivative of y(x), consider the inverse of y(x) (i.e. apply ln to both sides):

ln(y(x)) = f(x)

Now differentiate both sides, using the chain rule on the left side:

1/y(x) * y'(x) = f'(x)

Then just multiply both sides by y(x) to solve for y'(x):

y'(x) = y(x)*f'(x) = exp(f(x)) * f'(x)


I'm not sure it is possible to approach this problem using the definition of a derivative. If you assumed that (exp(x))' = exp(x), you could possibly expand exp(x) in a Taylor series and find the limit that way, but I wouldn't consider that a rigorous proof.

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Thanks for the replys guys, thanks for the info on f(x+hx) = hf(x)+f(x), did not know it doesnt apply in all cases. Thanks for the proof too Chaotic_Attractor, I am already familar with that proof but thank for the post. I quess the better question now is why can one not start with the quoitent quotient, what is the mathematical reason that doesnt allow the differenciation of e^f(x) starting from there?

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Question 1: assuming that the notation δf(x) means, as per the general assumption, the same as f(x + δx) - f(x), then your step iv is correct by virtue of the equality f(x + δx) - f(x) + f(x) = f(x + δx) being a tautology. As such, Alrecenk is incorrect.

Your problem comes from step ix, which is a non-sequitur: there is no rule which allows you to replace a subexpression with its own limit. In practice, consider the limit of δx/δx as δx converges to zero: by replacing the top expression with its limit, you get 0, yet it is fairly obvious that the correct limit is 1. The same happens here (and your step x is incorrect, since the limit of (1-1) times anything is zero).

Question 2: you wonder why δuδv/δx has a limit of zero. The reason is simple: δu/δx has a limit of u', and δv/δx has a limit of v'. Therefore, δuδv/(δx)² has a limit of u'v'. And so, δx * δuδv/(δx)² has a limit of zero.

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Thanks for the reply ToohrVyk, make total sense although I would have never figured that out for myself searching round the net. Makes perfect sense, many thanks! Taking everything everyone has said on board ill try to reconsider my approach!

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I see now that Alrecenk incorrectly assessed what you were doing in that step, and I based my observation on that.

Check out this page:

http://everything2.com/index.pl?node_id=1417728

It will tell you how to use the limit quotient definition of the derivative to complete the proof.

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Folks, just a reminder about the school/homework policy on the Forum FAQ. Problems like this look academic and that is how I have to judge them. The OP here did show his work, which is the only reason I'm letting this thread stay open. As moderator, I have the authority to close this sort of thread if something seems inappropriate.

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grhodes_at_work, Thank you for keeping the thread open, I do appoligise if you may have thought this was homework. I can only give you my word that it isnt, it is for my own personal interest, I have posted before with regards to understanding the concepts of sets and notation used etc if you require proof of my intentions to simply learn about maths which I would be happy to pass to you! Thanks again!

and Chaotic_Attractor thanks for the post, you know I have never actually learn at school that e = lim n--> inf...etc. And that make everything so much easier, I suppose I just really wasnt looking basic enough in my working, the flaw with my method is that fact that I dont also consider e as a maniputable entity!

[Edited by - Genesiiis on April 21, 2008 5:20:46 PM]

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