\[\operatorname{Ai}'\left(z\right)=\operatorname{Ai}'\left(0\right)\left(1+\frac%
{2}{3!}z^{3}+\frac{2\cdot 5}{6!}z^{6}+\frac{2\cdot 5\cdot 8}{9!}z^{9}+\cdots%
\right)+\operatorname{Ai}\left(0\right)\left(\frac{1}{2!}z^{2}+\frac{1\cdot 4}%
{5!}z^{5}+\frac{1\cdot 4\cdot 7}{8!}z^{8}+\cdots\right),\]