\begin{align*} \frac{d}{dz}\left(z\uparrow^{2}n\right) & =\frac{d}{dz}\left(z\uparrow z\uparrow^{2}\left(n-1\right)\right)\\ & =\frac{d}{dz}\left(z^{z\uparrow^{2}\left(n-1\right)}\right)\\ & =z\uparrow^{2}\left(n-1\right)z^{z\uparrow^{2}\left(n-1\right)-1}+\log z\cdot z^{z\uparrow^{2}\left(n-1\right)}\frac{d}{dz}z\uparrow^{2}\left(n-1\right)\\ & =z\uparrow^{2}\left(n-1\right)\cdot z\uparrow^{2}n\cdot z^{-1}+\log z\cdot z\uparrow^{2}n\frac{d}{dz}z\uparrow^{2}\left(n-1\right)\\ & =\log^{n}z\left(\prod_{j=1}^{n}z\uparrow^{2}j\right)\sum_{k=1}^{n}\left\{ \left(\prod_{j=1}^{k}\frac{1}{\log z\cdot z\uparrow^{2}j}\right)\frac{d}{dz}\left(z\uparrow^{2}k\right)-\left(\prod_{j=1}^{k-1}\frac{1}{\log z\cdot z\uparrow^{2}j}\right)\frac{d}{dz}z\uparrow^{2}\left(k-1\right)\right\} \\ & =\log^{n}z\left(\prod_{j=1}^{n}z\uparrow^{2}j\right)\sum_{k=1}^{n}\left\{ \left(\prod_{j=1}^{k}\frac{1}{\log z\cdot z\uparrow^{2}j}\right)z\uparrow^{2}\left(k-1\right)\cdot z\uparrow^{2}k\cdot z^{-1}\right\} \\ & =\log^{n}z\left(\prod_{j=1}^{n}z\uparrow^{2}j\right)\sum_{k=1}^{n}\left\{ \frac{1}{\log^{k}z}\left(\prod_{j=1}^{k-2}\frac{1}{z\uparrow^{2}j}\right)\cdot z^{-1}\right\} \\ & =\frac{1}{z}\sum_{k=1}^{n}\left(\log^{n-k}z\right)\left(\prod_{j=k-1}^{n}z\uparrow^{2}j\right)\\ & =\frac{1}{z}\sum_{k=0}^{n-1}\left(\log^{k}z\right)\left(\prod_{j=n-k-1}^{n}z\uparrow^{2}j\right)\\ & =\frac{1}{z}\sum_{k=1}^{n}\left(\log^{k-1}z\right)\left(\prod_{j=n-k}^{n}z\uparrow^{2}j\right) \end{align*}