f'(x)=\displaystyle\sum^{+\infty}_{n=1}(-1)^{n-1}x^{3n-3}=\displaystyle\sum^{+\infty}_{n=1}(-1)^{n-1}\frac{x^{3n}}{x^3}=\frac{1}{x^3}\displaystyle\sum^{+\infty}_{n=1}(-1)^{n-1}x^{3n}=\frac{1}{x^3}\displaystyle\sum^{+\infty}_{n=1}(-1)(-1)^{n}(x^{3})^{n}=\frac{1}{x^3}\displaystyle\sum^{+\infty}_{n=1}(-1)(-x^{3})^{n}