Can the Riemann Zeta derivative be expressed in terms of Riemann Zeta? From this question: https://mathoverflow.net/questions/134600/zetax-in-terms-of-zetax-zeta1-x-gamma-psi it seems that Zeta can be expressed through its derivative:
$$\zeta(1-x) = \frac{2(\zeta'(x)\Gamma(x/2)+\Gamma((1-x)/2) \zeta'(1-x)\pi^{x-1/2})  )}{\Gamma((1-x)/2) \pi^{-1/2+x}(2\log\pi -\psi((1-x)/2)-\psi(x/2))}$$
I wonder, if the opposite is possible? Can the Riemann Zeta derivative be expressed in terms of Riemann Zeta? 
 A: With $\zeta(s):=\chi(1-s)\,\zeta(1-s), \, \, \chi(s):= \Gamma  \left( s \right) \cos \left(\frac12\pi s \right) 2 \left( 2\pi \right) ^{-s}$
the formula above can be simplified into:
$$\zeta(s):= -\dfrac{\zeta'(1-s)+\chi(s)\,\zeta'(s)}{\chi'(s)}$$
* begin addition *
There actually is a way to express $\zeta'(s)$ in terms of $\zeta(s)$ through the following formula:
$$\zeta'(s)= -\zeta(s)\cdot\big(\gamma + \Psi^{(0)}(1-s)\big) +\Psi^{(s-1)}(1)\cdot\Gamma(1-s)$$
with $\Psi^{(x)}(z)$ being the Polygamma function and $\gamma$ the Euler-Mascheroni constant.
However, there is an ugly snag here, since the analytic continuation of $\Psi^{(s-1)}(1)$ to complex and negative-integer values of $s$, is defined in terms of $\zeta'(s)$ and $\zeta(s)$... 
* end addition *
To give a (very partial) answer to your question, for $s=-2k, k=1,2,3...$ it is known that $\zeta'(s)$ can be expressed in terms of $\zeta(s)$ and a ratio involving $\pi^{2k}$ e.g.:
$\zeta'(-2)= -\dfrac{1}{4}\dfrac{\zeta(3)}{\pi^2}$
$\zeta'(-4)= \dfrac{3}{4}\dfrac{\zeta(5)}{\pi^4}$
$\zeta'(-6)= -\dfrac{45}{8}\dfrac{\zeta(7)}{\pi^6}$
$\dots$
The other likely candidate is $s=2$ where we have $\zeta'(2)= \zeta(2)\,\big(\gamma + \ln(2\,\pi) - 12\, \ln(A) \big)$ where $A$ is the Glaisher-Kinkelin Constant. However, since $A$ has a closed form in terms of $\zeta'(1)$ that leads to: $\zeta(2)\,\big(\gamma + \ln(2\,\pi) - 1 + 12\, \zeta'(-1) \big)$, I like to disregard this one as a valid solution.  
Could not find a more generic expressions for $\zeta'(s)$ and think the key problem is that no functional equation is known that links $\zeta'(s)$ to $\zeta'(1-s)$ that allows eliminating one of them.
