Help with differentiation incl. ln.

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December 18th 2011, 03:59 PM
SimonFrisendal

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Help with differentiation incl. ln.

Hello MathHelp.
So i'm trying to solve this problem below
Attachment 23105
And using the rules of differentiation that i know of i fail. Perhaps the book that i am reading is wrong. If anyone can help it would be greatly appreciated.

- Simon.
December 18th 2011, 06:37 PM
vincisonfire

Re: Help with differentiation incl. ln.

Hi,
These is nothing special there. Just use the product rule together with the chain rule.
Note that you have something like $(x-1)ln(x-1)-x ln(x)$ . If you differentiate you will have something of the form $ln(x-1)-{x-1\over x-1}-ln(x)+{x\over x} = ln(x-1)-ln(x)$ .
It is true that all the constants makes it messy, but keep you head straight and just use the basics rules you know.
December 18th 2011, 06:46 PM
pickslides

Re: Help with differentiation incl. ln.

First take out the constant

$\displaystyle \frac{d}{dE}\kappa_B\left( \left(1+\frac{E}{\epsilon}\right)\times \ln\left(1+ \frac{E}{\epsilon}\right)-\frac{E}{\epsilon}\ln\frac{E}{\epsilon}\right)$

$\displaystyle = \kappa_B\frac{d}{dE}\left( \left(1+\frac{E}{\epsilon}\right)\times \ln\left(1+ \frac{E}{\epsilon}\right)-\frac{E}{\epsilon}\ln\frac{E}{\epsilon}\right)$

then try the product rule term by term i.e.

$\displaystyle \frac{d}{dE}\left( \left(1+\frac{E}{\epsilon}\right)\times \ln\left(1+ \frac{E}{\epsilon}\right)\right)$

$\displaystyle = \frac{d}{dE}\left(1+\frac{E}{\epsilon}\right) \times \ln\left(1+ \frac{E}{\epsilon}\right)+\left(1+\frac{E}{\epsilo n}\right)\times \frac{d}{dE}\ln\left(1+ \frac{E}{\epsilon}\right)$
December 19th 2011, 12:05 AM
SimonFrisendal

Re: Help with differentiation incl. ln.

Thank you pickslides :) That's easier than expected.

- Simon.