Hi,
These is nothing special there. Just use the product rule together with the chain rule.
Note that you have something like $\displaystyle (x-1)ln(x-1)-x ln(x)$. If you differentiate you will have something of the form $\displaystyle ln(x-1)-{x-1\over x-1}-ln(x)+{x\over x} = ln(x-1)-ln(x)$.
It is true that all the constants makes it messy, but keep you head straight and just use the basics rules you know.
First take out the constant
$\displaystyle \displaystyle \frac{d}{dE}\kappa_B\left( \left(1+\frac{E}{\epsilon}\right)\times \ln\left(1+ \frac{E}{\epsilon}\right)-\frac{E}{\epsilon}\ln\frac{E}{\epsilon}\right)$
$\displaystyle \displaystyle = \kappa_B\frac{d}{dE}\left( \left(1+\frac{E}{\epsilon}\right)\times \ln\left(1+ \frac{E}{\epsilon}\right)-\frac{E}{\epsilon}\ln\frac{E}{\epsilon}\right)$
then try the product rule term by term i.e.
$\displaystyle \displaystyle \frac{d}{dE}\left( \left(1+\frac{E}{\epsilon}\right)\times \ln\left(1+ \frac{E}{\epsilon}\right)\right) $
$\displaystyle \displaystyle = \frac{d}{dE}\left(1+\frac{E}{\epsilon}\right) \times \ln\left(1+ \frac{E}{\epsilon}\right)+\left(1+\frac{E}{\epsilo n}\right)\times \frac{d}{dE}\ln\left(1+ \frac{E}{\epsilon}\right)$