Doubt on Derivation of beta function of QED from Ryder's book on QFT page 345 I'm studying 1 loop renormalization  of QED using QFT by Ryder.
On page 345,
$$e_B=e\mu ^ {\epsilon \over 2} \Bigg(1+{e^2 \over {12\pi ^2 \epsilon}}\Bigg).$$
differentiating the above equation  gives, to order $e^3$
$$\mu {\partial e \over \partial \mu }=-{\epsilon \over 2}e+{e^3 \over 12\pi ^2}.$$ 
I don't get this equation What I get is
$${\partial e_B \over \partial \mu} ={e\mu^{{\epsilon \over2 }-1}\over 2}\Bigg(\epsilon +{e^2\over 12 \pi^2}\Bigg)   +\mu^{\epsilon\over 2}{\partial e\over \partial \mu }\Bigg(1+{e^2 \over  4\pi^2\epsilon} \Bigg).$$
I don't find any connection  between this two equations can any one please help, Your little time may save my lot of time.
 A: The elephant in the room is the following question:
How can we treat $\frac{e^2}{4\pi^2\epsilon}$ as subleading as compared to $1$? In dimensional regularization the parameter $\epsilon$ is supposed to be small. 
The brief answer is that renormalization is first-and-foremost a perturbative formal power series in the coupling constant $e^2$. Secondly, each coefficient of this formal power series is a truncated Laurent series in $\epsilon$. Not the other way around.
A: As per @Qmechanic's suggestion, you should have gotten 
$$
0= {\epsilon \over 2} e  \Bigg(1+{e^2 \over {12\pi ^2 \epsilon}}\Bigg)+\mu {\partial e \over \partial \mu }  \Bigg(1+{3e^2 \over {12\pi ^2 \epsilon}}\Bigg),  $$
i.e.
$$\mu {\partial e \over \partial \mu }=-{\epsilon \over 2}e \frac{1+{e^2 \over {12\pi ^2 \epsilon}}}{1+{3e^2 \over {12\pi ^2 \epsilon}}}\\
=-{\epsilon \over 2}e  \Bigg(1+{e^2 \over {12\pi ^2 \epsilon}}-{3e^2 \over {12\pi ^2 \epsilon}} + O(e^4)\Bigg)=
-{\epsilon \over 2}e+{e^3 \over 12\pi ^2}+ O(e^5).$$ 
You may then take $\epsilon \to 0$ "safely". The coupling then increases with energy. 
