The main goal of this thesis is to study the problem of numerical derivative computation of the complex and real functions of a real argument. The introductory chapter gives an overview of the basic methods for numerical derivative computation. In the second chapter, the algorithm for numerical derivative computation of the complex and real functions of a real argument using the Taylor formula, i.e. by differentiation, is described. Numerical derivatives of a function can be computed using forward, backward and central numerical algorithms, of which central algorithm gives the best approximation of the analytical function derivative. In the third chapter, an algorithm for numerical derivative computation of the complex functions of a complex argument using Cauchy integral formula is described. Cauchy integral formula in combination with Gauss-Legendre numerical integration can be used to numerically compute the derivatives of complex analytic functions of a complex argument, where complex and real analytic functions of a real argument are a special case. This method avoids most strong mathematical requirements, and is implemented by numerical integration along a curve closed around the point at which derivative is computed. The fourth chapter describes a numerical algorithm for computing the first derivative of real functions using complex-step derivative approximation. The main advantage of this method is that for a very small step it has a very high accuracy, and its accuracy is limited only by the accuracy of the compiler. This is the simplest method for computing the first derivative of real functions. In the fifth chapter, examples of numerical computation of the first and higher derivatives of the selected real function are given. A comparison of the obtained results with the analytical derivative, as expected, showed that it is best to use a numerical algorithm with a complex step to compute the first numerical derivative of a real function. Cauchy’s integral formula combined with Gauss-Legendre numerical integration can be used to numerically compute the derivatives of complex analytic functions of a complex argument, where complex and real analytic functions of a real argument are only a special case. This method avoids most stringent mathematical requirements, and is reduced to numerical integration along a curve closed around the point where derivative is computed.