It is useful to apply interval estimates to improve the evaluation of reliability results of calculations, and therefore the evaluation of the reliability of mechanical structures. In this paper, interval estimates are used to establish the range of variation of a function and its derivatives As is known, the problem of the simultaneous approximation of a function and its derivatives cannot be solved using classical interpolation polynomials. In this paper, we consider the approximation of a function and its first derivative by using polynomial and trigonometric splines with the third order of approximation. In this case, the approximation of the first derivative turns out to be discontinuous at the nodes of the grid. The values of the constants in the estimates of the errors of approximation with the trigonometric and polynomial splines of the third order are given. It is shown that these constants cannot be reduced. To solve practical problems, it is often important not to calculate the values of the function and its derivatives in a number of nodes on the grid interval, but to estimate the range of change of the function on this interval. For the interval estimation of the approximation of function or its first derivative, we use the technique of working with real intervals from interval analysis. The algorithms for constructing the variation domain of the approximation of the function and the first derivative of this function are described. The results of the numerical experiments are given.