We determine how to obtain periods, quasi-periods, logarithms and quasi-logarithms of uniformizable abelian Anderson t-modules as evaluations of rigid analytic trivializations of associated dual t-motives. Building on this and utilizing properties of hyperderivatives, we show that hyperderivatives of periods and logarithms of uniformizable abelian Anderson t-modules can be realized as evaluations of rigid analytic trivializations of Maurischat's prolongations of the associated dual t-motives. If time permits, we will further discuss these results using particular examples such as Drinfeld modules, n-th tensor power of Carlitz module, etc. This is joint work with Matt Papanikolas.