  We investigate periods, quasi-periods, logarithms, and quasi-logarithms of
Anderson $t$-modules, as well as their hyperderivatives. We develop a
comprehensive account of how these values can be obtained through rigid
analytic trivializations of abelian and $\mathbf{A}$-finite $t$-modules. To do
this we build on the exponentiation theorem of Anderson and investigate
quasi-periodic extensions of $t$-modules through Anderson generating functions.
By applying these results to prolongation $t$-modules of Maurischat, we
integrate hyperderivatives of these values together with previous work of
Brownawell and Denis in this framework.
