A view of dynamic derivatives on time scales from approximations.

There has been a considerable number of recent discussions on applications of different types of dynamic derivatives. The dynamic derivatives not only play a central role in time scales theory and methods, but also provide a remarkable way of approximations in mathematical modelling and computational applications. In this paper, we will explore connections between existing dynamic and conventional derivatives over the time scales used. We will show that while first order dynamic derivatives provide acceptable accuracies in approximating conventional derivatives, second order dynamic derivatives are in general inconsistent with conventional derivatives. It is evident that, before incorporating structures of particular time scales used into the dynamic derivative formulae properly, one needs to be prudential when bridging discrepancies between the differential and difference equations modelling similar applications by higher order dynamic derivatives. Numerical experiments are given to further illustrate our conclusions. [ABSTRACT FROM AUTHOR]