Two theorems we will use about uniform convergence:
Theorem 1 Let $f,f_n:X\to \mathbb{R}$. If $f_n$ is continuous at $a\in X$ for $n=0,1,...$ and $\sum_{k=0}^{\infty}f_k$ converges uniformly then $\sum_{k=0}^{\infty}f_k(x)$ is continuous at $a$
Theorem 2 Let $f_n:[a,b]\to \mathbb{R}$. If $f_n$ is integrable for $n=0,1,...$ and $\sum_{k=0}^{\infty}f_k$ converges uniformly then \begin{equation}\int_{a}^{b}\sum_{k=0}^{\infty}f_k=\sum_{k=0}^{\infty}\int_{a}^{b}f_k \end{equation}
Now as the OP said the series $\sum_{n=1}^{\infty}\frac{x^n}{n^2}$ converges uniformly in $[-1,1]$ and to a function $f:[-1,1]\to \mathbb{R}$. The same can be said for the derived series $\sum_{n=1}^{\infty}\frac{x^{n-1}}{n}$ which converges unifomly in $(-1,1)$ to some $g$. That $g$ is continuous and so \begin{equation}\int_{a}^{x}g=\sum_{n=1}^{\infty}\int_{a}^{x}\frac{t^{n-1}}{n}\, dt= \sum_{n=1}^{\infty}\frac{x^n}{n^2}-\frac{a^n}{n^2}=f(x)-\sum_{n=1}^{\infty}\frac{a^n}{n^2}\end{equation} for $-1. Therefore, by the 1st fundumental theorem of calculus, $f$ is differentiable in $(-1,1)$ and $f^{\prime}=g$.
Since $f$ is continuous in $[-1,1]$ and differentiable in $(-1,1)$, by the Mean Value Theorem for arbitrary small $h>0$, $\exists \xi\in (-1,-1+h):f^{\prime}(\xi)=\frac{f(-1+h)-f(-1)}{h} $ As $h\to 0^+$, $\xi \to -1^+$ and $f(x)\to \sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n}$ which converges and so $f$ is differentiable at $-1$.
Because $\lim_{x\to 1^-}f^{\prime}(x)=\sum_{n=1}^{\infty}\frac{1}{n}=+\infty$, $f$ is not differentiable at $1$. Remember that $f^{\prime}$ is only allowed to have essential discontinuities and not poles.
Therefore, $f$ is differentiable only in $[-1,1)$ and $f^{\prime}(x)=\sum_{n=1}^{\infty}\frac{x^{n-1}}{n}$