I am trying to differentiate this with respect to $\eta$ but am having some difficulty.
\begin{equation} g_0(1,\eta) = \frac{\frac{3\eta}{(\eta_c - \eta)} + \sum\limits_{k=1}^{4}kA_k\left(\frac{\eta}{\eta_c}\right)^k}{4\eta} \end{equation}
\begin{equation} g_0'(1,\eta) = \frac{(4\eta)\left(\frac{3\eta_c}{(\eta_c - \eta)^2}+\sum\limits_{k=1}^{4}k^2A_k\left(\frac{\eta}{\eta_c}\right)^{k-1}\right)-\left(\frac{3\eta}{(\eta_c - \eta)} + \sum\limits_{k=1}^{4}kA_k\left(\frac{\eta}{\eta_c}\right)^k\right)(4)}{(4\eta)^2} \end{equation}
Does this look right? When I continue from here, it just does not seem like anything is cancelling out or anything like that, so I want to make sure what I am doing looks correct.
\begin{eqnarray} \require{cancel} \frac{{\rm d}}{{\rm d}\eta}g(1, \eta) &=& \frac{1}{4}\frac{{\rm d}}{{\rm d}\eta} \left[ \frac{3\cancel{\eta}}{\cancel{\eta}(\eta_c - \eta)} + \sum_{k=1}^4 k A_k \frac{\eta^k}{\eta \eta_c^k}\right] \\ &=& \frac{3}{4} \frac{{\rm d}}{{\rm d}\eta} \frac{1}{\eta_c - \eta} + \frac{1}{4}\sum_{k=1}^4 \frac{k A_k}{\eta_c^k} \frac{{\rm d}\eta^{k-1}}{{\rm d}\eta} \\ &=& \frac{3}{4}\frac{1}{(\eta_c - \eta)^2} + \frac{1}{4}\sum_{k=1}^4\frac{k(k-1) A_k}{\eta_c^k}\eta^{k - 2} \\ &=& \frac{3}{4}\frac{1}{(\eta_c - \eta)^2} + \frac{1}{4}\sum_{k=1}^4\frac{k(k-1) A_k}{\eta_c^2}\left(\frac{\eta}{\eta_c}\right)^{k - 2} \end{eqnarray}