Derivative of Riemann Zeta Function

Theorem
The derivative of the Riemann zeta function is

$$\frac{d\zeta}{dz} = \sum_{n=1}^\infty \frac{\log(n)}{n^z} \ $$

Proof
$$\frac{d\zeta}{dz} = \frac{d}{dz} \left({ \sum_{n=1}^\infty \frac{1}{n^z}  }\right) = \sum_{n=1}^\infty \frac{d}{dz} \left({ n^{-z}  }\right)  = \sum_{n=1}^\infty  \left({ \log(n) n^{-z} }\right) = \sum_{n=1}^\infty \frac{\log(n)}{n^z} \ $$