On the derived series of some groups

We solve Problems 17.82 and 17.86(b) posed by Mikhailov in the Kourovka Notebook [1]. Namely, we construct: (1) an example of a finitely presented group $H$ in which the intersection $H^{(\omega)}$ of all terms of the derived series is distinct from its commutant; (2) an example of a balanced presentation $\langle x_1,x_2,x_3\mid r_1,r_2,r_3\rangle$ of the trivial group for which $F(x_1,x_2,x_3)/[R_1,R_2]$ is not a residually soluble group (here $R_i$ ($i=1,2$) denotes the normal closure of $r_i$ in $F(x_1,x_2,x_3)$). The construction of the second example is related to some approach to the Whitehead asphericity conjecture.