  The explicit formula for the hyperbolic metric
$\lambda_{\alpha,\,\beta,\,\gamma}(z)|dz|$ on the thrice-punctured sphere
$\mathbb{P} \backslash \{z_1,\,z_2,\,z_3\}$ with singularities of order
$\alpha,\,\beta,\,\gamma \leq 1$ with $\alpha+\beta+\gamma>2$ at
$z_1,\,z_2,\,z_3$ was given by Kraus, Roth and Sugawa in \cite{Rothhyper}. In
this paper we investigate the asymptotic properties of the higher order
derivatives of $\lambda_{\alpha,\,\beta,\,\gamma}(z)$ near the singularity and
give some more precise description for the asymptotic behavior.
