A plethora of methods are used for solving equations in the finite-dimensional Euclidean space. Higher-order derivatives, on the other hand, are utilized in the calculation of the local convergence order. However, these derivatives are not on the methods. Moreover, no bounds on the error and uniqueness information for the solution are given either. Thus, the advantages of these methods are restricted in their application to equations with operators that are sufficiently many times differentiable. These limitations motivate us to write this paper. In particular, we present the more interesting semi-local convergence analysis not given previously for two sixth-order methods that are run under the same set of conditions. The technique is based on the first derivative that only appears in the methods. This way, these methods are more applicable for addressing equations and in the more general setting of Banach space-valued operators. Hence, the applicability is extended for these methods. This is the novelty of the paper. The same technique can be used in other methods. Finally, examples are used to test the convergence of the methods.