  The article examines Nikolskii and Besov spaces with norms defined using
"$L_p$-averaged" mixed moduli of continuity for functions of appropriate
orders, instead of mixed moduli of continuity of known orders for certain mixed
derivative functions. The work provides upper and lower estimates of the best
accuracy of reconstruction derivatives from function values at a given number
of points for such classes of functions in bounded domains of a certain kind.
These estimates are not weaker, but in some cases even stronger than those
derived by the author in the problem under consideration for the aforementioned
classes of functions on cube $ I^d. $ It also significantly broadens the class
of Nikolskii and Besov spaces of mixed smoothness for which mentioned estimates
in the problem under consideration have been derived.
