Calculation of $n$th derivative with minimum error based on function’s measurement

A solution of the problem that arises in all cases where it is required to approximately calculate the derivatives of an a priori smooth function by its experimental discrete values is proposed. The problem is reduced to finding an “optimal” step of difference approximation. This problem has been studied by many mathematicians. It turned out that to find an optimal approximation step for the $k$th-order derivative, it is required to know a highly accurate estimate of the modulus of the derivative of order $k+1$. The proposed algorithm, which gives such an estimate, is applied to the problem of thrombin concentration, which determines the dynamics of blood coagulation. This dynamics is represented by plots and provides a solution of the thrombin concentration problem, which is of interest to biophysicists.