Sommerfeld Expansion & Chemical Potential

[RicardoMP]

Hi!
I'm trying to show how the chemical potential depends on the temperature and I'm advised to use the Sommerfeld expansion. I'm using it on the density of charge $$n=\int^{+\infty}_{-\infty} \rho(\epsilon)n_Fd\epsilon$$, which gives $$n=\int^{\mu}_{0} \rho(\epsilon)d\epsilon +\frac{\pi^2}{6}(k_BT)^2\frac{d\rho(\epsilon)}{d\epsilon}|_{E=\mu}$$.
After having a hard time trying to insert a term with a $$\mu$$ in it, I searched on google and found the following excerpt that I uploaded. I don't understand the second paragraph and how it gives the $$(\mu-E_F)\rho(\epsilon)$$ term.
Thank you for your time.

 

[RicardoMP]

Hi!
I'm trying to show how the chemical potential depends on the temperature and I'm advised to use the Sommerfeld expansion. I'm using it on the density of charge $$n=\int^{+\infty}_{-\infty} \rho(\epsilon)n_Fd\epsilon$$, which gives $$n=\int^{\mu}_{0} \rho(\epsilon)d\epsilon +\frac{\pi^2}{6}(k_BT)^2\frac{d\rho(\epsilon)}{d\epsilon}|_{E=\mu}$$.
After having a hard time trying to insert a term with a $$\mu$$ in it, I searched on google and found the following excerpt that I uploaded. I don't understand the second paragraph and how it gives the $$(\mu-E_F)\rho(\epsilon)$$ term.
Thank you for your time.

Nevermind! Taylor Series, duh! xD