If

$\displaystyle \displaystyle f(x)= \sum_{n=0}^{\infty} \frac{(-1)^nx^{2n+1}}{2n+1}$,

calculate $\displaystyle f'(x)$ and use it to find $\displaystyle f(x)$ explicitly. Justify each of your steps.

so I have obtained $\displaystyle f'(x)= (-1)^nx^{2n}$, what is the next step?