Mortality Table: Derivation of the Variance of q_j

Can somebody quickly go through both of these for the exact/actuarial methods? I tried to derive them myself via the delta method (at least for the exact method) but couldn't quite get it.



I missed a couple of questions regarding this in the past practice exams that I couldn't get by just memorizing the formulas.. so I'd at least like to know conceptually how the variances came to be.



Thanks.

Mortality Table: Derivation of the Variance of q_j