In the framework of the cascade approach to state observers design for dynamic objects under the influence of external disturbances, a method for reconstructing the derivatives of a deterministic time function from its current values is proposed, which does not require knowledge of the analytical form of the function and numerical differentiation. Assuming that the function is piecewise smooth and its derivatives are bounded by known constants, a virtual dynamic model of canonical form with unknown input is introduced. Based on this model, whose order depends on the order of the derivatives to be restored, a dynamic differentiator is constructed in the form of a state observer with piecewise linear control actions. On the basis of sufficient stability conditions, inequalities are obtained for adjusting the parameters at which the observer’s variables converge to unknown derivatives with a given accuracy for a given time. In this paper, these constructions are demonstrated on the example of a control system for a wheeled robot. Current information about the state variables of the control plant, given actions and their derivatives up to the second order is required to implement feedback in the path following problem. It is assumed that the reference signals enter the control system in real time from an autonomous source and there is no analytical description. A dynamic differentiator is used in the feedback loop to estimate the derivatives of the reference signals. The efficiency of the developed approach is confirmed by the results of numerical simulation.