For the derivatives $f^{(k)}_\alpha(x)$ of the one-sided stable density of index $\alpha \in (0, 1)$ asymptotic formulas are computed as $k \rightarrow \infty$ thereby exhibiting the detailed analytic structure for large orders of derivatives. The results extend those for the well-known case $\alpha = \frac{1}{2}$ which may be expressed in terms of Laguerre polynomials (formulas of Plancherel-Rotach type).