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We characterize the marked bordered unpunctured oriented surfaces with theproperty that all the Jacobian algebras of the quivers with potentials arisingfrom their triangulations are derived equivalent. These are either surfaces ofgenus g with b boundary components and one marked point on each component, orthe disc with 4 or 5 points on its boundary.  We show that for each such marked surface, all the quivers in the mutationclass have the same number of arrows, and the corresponding Jacobian algebrasconstitute a complete derived equivalence class of finite-dimensional algebraswhose members are connected by sequences of Brenner-Butler tilts. In addition,we provide explicit quivers for each of these classes.  We consider also 10 of the 11 exceptional finite mutation classes of quiversnot arising from triangulations of marked surfaces excluding the one of thequiver X_7, and show that all the finite-dimensional Jacobian algebras in suchclass (for suitable choice of potentials) are derived equivalent only for theclasses of the quivers E_6^(1,1) and X_6.

Mutation classes of certain quivers with potentials as derived equivalence classes