On the derivation of field equation of f (R) gravity with kinetic scalar curvature

(In given work we show details of variation of the f (R) gravity action in the metric formalism; we demonstrate Gibbons - York - Hawking method of a boundary integral removing. We present the derivation of fields equations of f (R) gravity in details. We derive the covariant field equations of f (R) gravity theory with the kinetic scalarcurvature term, i.e. the model which based on the action integral: S = r d 4 xv-g R, j)R, j X (R) + f (R + Smat t er.