The explicit formula for the hyperbolic metric ${lambda}_{{alpha},{eta},{gamma}}(z){mid}dz{mid}$ on the thrice-punctured sphere $mathbb{P}{ackslash}{0,1,{infty}}$ with singularities of order 0 < ${alpha}$, ${eta}$ < 1, ${gamma}{leq}1$, ${alpha}+{eta}+{gamma}$ > 2 at 0, 1, ${infty}$ was given by Kraus, Roth and Sugawa in [9]. In this article we investigate the asymptotic properties of the higher order derivatives of ${lambda}_{{alpha},{eta},{gamma}}(z)$ near the origin and give more precise descriptions for the asymptotic behavior of ${lambda}_{{alpha},{eta},{gamma}}(z)$.