Several schemes of iterative algorithms for calculating a simple root of a nonlinear equation are proposed in the article. All algorithms are based on the use of the $q$-derivative $$ D_qf(u)=\frac{f(qu)-f(u)}{qu-u}, \quad q\ne 1. $$ The authors perform a convergence analysis of the methods and compare them with classical methods. Numerous computational results and numerical comparisons between different algorithms are given.