The limit state function in the reliability analysis of geotechnical engineering is often very complicated or even implicit

so it’s difficult to calculate the limit state function’s gradients analytically. However

the solution of the gradients is necessary and important in some general used reliability analysis methods

such as the first order reliability method(FORM) and the second order reliability method(SORM). Therefore

it’s necessary to solve the gradients of the limit state function by numerical methods. The authors studied carefully two numerical differential methods

finite difference method(FDM) and rational polynomial technique(RPT)

and pointed that FDM is the special case of RPT when the limit state function is linear. Accordingly

FDM3 can be used for the continuous linear limit state function and RPT5 for the continuous nonlinear limit state function. For the discontinuous limit state function

RPT should be used because of the large error of FDM. In each case

the control coefficient of step size is suggested to be 1.