On the derivative of the Minkowski question mark function $?(x)$

Let $x=[0;a_1,a_2,\mathrm{.}\mathrm{.}\mathrm{.}]$ be the decomposition of the irrational number $x \in [0,1]$ into regular continued fraction. Then for the derivative of the Minkowski function $?(x)$ we prove that ${?}^{\mathrm{\prime }}(x)=+\mathrm{\infty }$ provided $ \limsup_{t\to \infty}\frac{a_1+…+a_t}{t} <\kappa_1 =\frac{2\log \lambda_1}{\log 2} = 1.388^+$,and$?‘(x) = 0$provided$ \liminf_{t\to \infty}\frac{a_1+…+a_t}{t} >\kappa_2 = \frac{4L_5-5L_4}{L_5-L_4}= 4.401^+$(here$ L_j = \log (\frac{j+\sqrt{j^2+4}}{2}) - j\cdot\frac{\log 2}{2}$). Constants ${\kappa }_1,{\kappa }_2$ are the best possible. Also we prove that $?’(x) = +\infty$holdsforall$x$ with partial quotients bounded by 4.