I am trying to understand the proof of the fact that the $L-\text{function}$ given by $L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^s}$ ($\chi$ is a Dirichlet character modulo $q$) is continuously differentiable for $0<s<1$. I am referring to chapter 8 of Stein and Shakarchi's Fourier Analysis for the proof. I understand why $L(s,\chi)$ is continuous but I am unable to show that it is continuously differentiable.
The proof begins with differentiating $f(s)=\sum_{n=1}^{N}\frac{\chi(n)}{n^s}$ term by term and the derivative is $-\sum (\log n) \frac{\chi(n)}{n^s}$. We can write $f'(s)$ as $\sum s_k[-k^{-s}\log k+(k+1)^{-s}(k+1)]$ where $s_k=\sum_{n=1}^{k}\chi(n)$
Now the authors consider $g(x)=x^{-s}\log x$ and state that each of the terms $s_k[-k^{-s}\log k+(k+1)^{-s}(k+1)]$ is $O(k^{-\delta/2-1})$ (for each $s>\delta>0$) but I have no idea why. All I can see is $g'(x)=x^{-s-1}(1-\log x)$ and applying the mean value theorem, each term is of the form $s_k(g(k+1)-g(k))=s_k(g'(c))$. By a previous theorem $|s_k|\leq q$ and $|x^{-s-1}|<k^{-s-1}<k^{-\delta -1} $, so $|s_k[-k^{-s}\log k+(k+1)^{-s}(k+1)]|= s_k(g(k+1)-g(k))<qk^{-\delta -1}(1-\log c)$ but I am stuck here.