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 Chapter 6 
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Joined: 03 Jul 2010, 14:49
Posts: 1
Post Chapter 6
I learnt some very basic calculus some 58 years ago! I am happy with finding the differential of say y = x**2 - by which I mean x squared - but I am completely thrown when it comes to finding a differential of y = 2**x and completely at sea when looking at the solution for 6.02 which gives the differential for e to the minus 1 over x. Can anyone point me to some website or text book where I can learn a bit more so as to understand it all?


07 Mar 2011, 18:19

Joined: 20 Jul 2011, 10:24
Posts: 1
Post Re: Chapter 6
Nicolas,

You might well have received an answer to this question, but anyway:

The approach to differentiating equations of this kind is to take logs, so:

y=2^x


becomes

ln y = ln (2^x)


or

ln y = x ln 2


Differentiating (implicitly)gives:

(1/y).(dy/dx) = ln 2


and finally

dy/dx = 2^x. ln 2


As regards the solution to 6.02 and the comment re. solutions of exp(-1/x): the method above will show that the first derivative is
(1/x^2).exp(-1/x)

and all subsequent derivatives will have the form as stated of
P(1/x).exp(-1/x)

where P(1/x) is a polynomial in 1/x.

Hope that helps.


20 Jul 2011, 14:24

Joined: 12 Jul 2010, 07:44
Posts: 154
Post Re: Chapter 6
Or you could note that one of the basic, well-known properties of e is that \frac{de^x}{dx} = e^x, and so by the chain rule:


\frac{d}{dx}e^{F(x)} = \frac{dF}{dx}.\frac{de^F}{dF} = \frac{dF(x)}{dx}.e^{F(x)}


To differentiate 2^x, you then just need to note that 2^x = e^{x\ln 2} (i.e. in this case F(x) = (\ln 2)x).


18 Aug 2012, 18:09

Joined: 11 Jul 2009, 20:45
Posts: 25
Post Re: Chapter 6
Nicolas ,

x**2 is well defined ie x times x . The problem with a number like e being multiplied ( EG e**x ), unknown times, is that the mathefuckstechians get really bent out of shape on this one . Then they go to the rules , or better yet show you ln (e**x ) =x .

the reason x**2 makes sense is because it is simple multipication ie, x*x , and we can do a small increment on that
( x +dx )*( x+dx ) = x*x + 2x*dx + dx*dx and to first order the change from x*x
is 2x times the dx change in x .

Don't let the priests of science change your noble ways Nicolas !


20 Jun 2013, 01:16

Joined: 12 Jul 2010, 07:44
Posts: 154
Post Re: Chapter 6
harried, I don't see how your response is in any way helpful to Nicolas Bellord, or anyone else on this board. It certainly won't help anyone solve exercise 6.02.

Looking through your other posts, it would seem that the bulk of them consist of incoherent, oddly-punctuated rants about how complex logarithms don't make sense to you... frequently with a generally resentful tone vaguely suggesting that it's because of some kind of conspiracy on the part of those "mathefuckstechians" to deliberately confuse us (or you).

Now personally I resent the insult: it is by implication a slur on the entire field of mathematics, the study of which (together with physics) is the whole reason the rest of us are reading RTR and posting on this board in the first place. So, gee, thanks for that.

I'm no expert, but it seems clear to me that you suffer from paranoid schizophrenia. (Either that, or perhaps we're being made the butt of some extended ongoing joke, and you've got an extremely bizarre sense of humour... but no, the first explanation seems far more likely).

In any case, given the attitude and opinions you've demonstrated, I don't see how either reading RTR, or posting about it here could possibly be a productive enterprise for you. I'd strongly urge you to see your doctor, take your medication, forget about maths altogether, and find a completely different hobby that's less evidently stressful!

However if you must post, please try to stay on-topic, and stick to content that's likely to be helpful or explanatory to the others of us on this particular board.

Thanks.


20 Jun 2013, 07:16
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