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The article examines Nikolskii and Besov spaces with norms defined using"$L_p$-averaged" mixed moduli of continuity for functions of appropriateorders, instead of mixed moduli of continuity of known orders for certain mixedderivative functions. The work provides upper and lower estimates of the bestaccuracy of reconstruction derivatives from function values at a given numberof 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 thosederived by the author in the problem under consideration for the aforementionedclasses of functions on cube $ I^d. $ It also significantly broadens the classof Nikolskii and Besov spaces of mixed smoothness for which mentioned estimatesin the problem under consideration have been derived.

Reconstructing derivatives from values of functions belonging to Nikolskii-Besov classes of mixed smoothness in domains of a certain kind