The paper describes the moving approximation algorithms for the functions, which have continuous and bounded derivatives of the first or higher orders. Firstly, Lagrange mean theorem is generalized for the equal and not equal steps. Additionally, la-grange mean theorem is generalized for the reduced time approximation. Estimations of the residuals in the generalized Lagrange theorems are proposed. Secondly, we consider application of the generalized Lagrange theorems for the design moving approximation algorithms. It is demonstrated, that an error approximation depends on the appropriate residual in the generalized la-grange theorems. Thirdly, we obtain results which allow us to compensate for an error approximation with a given accuracy. This fact is achieved due to a feedback compensation for the error approximation by using the derivative observers. The values of the time approximation and estimates of the approximation errors are presented. Simulations demonstrate that an approximation of the smooth functions by using algorithms with a compensation for the approximation error is better than an approximation without a compensation for the approximation error. If an approximated function has discontinuities in derivatives, it is recommended to use the algorithms without approximation with an error compensation, since the value of the function at the output of the observer in the derivative points of the discontinuity can be quite large.