In the present paper, having a look at the anomalous diffusion phenomena, by extending the concept of fractional differential operator of Caputo-Fabrizio type, to include a class of variable orders, a time-scale counterpart of the non-integer order differential operator is introduced. Implementing the measure theory on time scales and introducing Dirac delta functional, based on the left density property of a given point, it has been deduced that, regarding left density property of a point, the both limit cases, namely, when the order is approaching 0 or 1, give the well-known delta-derivative and a shifted zero-th derivative respectively. By making use of the Laplace transform and direct delta-calculus based approach, the associated integral for constant and variable orders are discussed. To illustrate the theory, some dynamic equations on time scales are studied and a dynamic problem, which governs a class of viscoelastic oscillation phenomena and involving the new introduced derivative is studied.