The paper discusses the improved least square method for polynomial differentiation. The method is suitable for real-time data processing and real-time control by a computer. It covers optimal estimations of both momentary value and continuous function curve of polynomial differential. Smoothing time may be taken at the sampling time or between any two neighboring samples

For middle-point smoothing

the number of samples may be odd or even. Simple formulas of weight coefficients and estimation errors are also shown in the paper. The law of error changes is easily shown by formulas of estimation errors

which is very useful for the correct use of the least square method.