A triangular system of integral equations is obtained, which, under certain conditions, are satisfied by the derivatives with respect to the parameters of the solution of an integral equation of the 2nd kind. On this basis, some algorithms of the Monte Carlo method are constructed and studied. The corresponding estimates of the first derivatives are in fact the same as the familiar scalar estimates, but the matrix approach greatly simplifies the study of the variances and extends the sufficient condition for them to be finite. Similar results are obtained for the parametric disturbances of the solution of an integral equation. Examples are given in which the method is used in the theory of radiation transport, where it turns out that, in certain cases, the new algorithm for the estimation of the small disturbance differs from the earlier algorithm based on the theory of small disturbances for the transport equation.