# Derivative of Riemann Zeta Function

$\displaystyle \map {\zeta'} z = \frac {\d \zeta} {\d z} = -\sum_{n \mathop = 2}^\infty \frac {\map \ln n} {n^z}$
 $\displaystyle \frac {\d \zeta} {\d z}$ $=$ $\displaystyle \map {\frac \d {\d z} } {\sum_{n \mathop = 1}^\infty n^{-z} }$ $(1):\quad$ $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop = 1}^\infty \map {\frac \d {\d z} } {n^{-z} }$ $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop = 1}^\infty \paren {-\map \ln n n^{-z} }$ Derivative of Exponential Function $\displaystyle$ $=$ $\displaystyle -\sum_{n \mathop = 1}^\infty \frac {\map \ln n} {n^z}$ $\displaystyle$ $=$ $\displaystyle -\sum_{n \mathop = 2}^\infty \frac {\map \ln n} {n^z}$ as $\ln 1 = 0$
$\blacksquare$