(1)

\begin{align*} Li_{s+1}'(z) & =\sum_{k=1}^{\infty}\frac{kz^{k-1}}{k^{s+1}}\\ & =\frac{1}{z}\sum_{k=1}^{\infty}\frac{z^{k}}{k^{s}}\\ & =\frac{Li_{s}(z)}{z} \end{align*}

(2)

\begin{align*} Li_{s+1}(z) & =\int_{0}^{z}Li_{s+1}'(t)dt+Li_{s+1}(0)\\ & =\int_{0}^{z}\frac{Li_{s}(t)}{t}dt \end{align*}

(2)-2

\begin{align*} Li_{s+1}(z) & =\sum_{k=1}^{\infty}\frac{z^{k}}{k^{s+1}}\\ & =\int_{0}^{z}\sum_{k=1}^{\infty}\frac{t^{k-1}}{k^{s}}dt\\ & =\int_{0}^{z}\frac{1}{t}\sum_{k=1}^{\infty}\frac{t^{k}}{k^{s}}dt\\ & =\int_{0}^{z}\frac{Li_{s}(t)}{t}dt \end{align*}