\[\operatorname{Ai}'\left(z\right)=-\tfrac{1}{2}\pi^{-1/2}z^{1/4}W_{0,2/3}\left(%
2\zeta\right)=-3^{1/6}\pi^{-1/2}\zeta^{4/3}e^{-\zeta}U\left(\tfrac{7}{6},%
\tfrac{7}{3},2\zeta\right),\]