The necessary and the sufficient conditions for the solution of the equality constrained optimization problem withNvariables and(N−1)constraints are first derived and generalized toNvariables andmconstraints. Directional derivatives are used in the approach. The necessary and sufficient conditions for the problem withN−1 constraints are shown to be equivalent to the unconstrained one-variable problem, when the ordinary derivatives are replaced by the corresponding directional derivatives of the objective function in the direction tangent to the intersection of the constraints. The general equality constrained optimization problem ofNvariables andmconstraints is then analysed using the directional derivative approach. Feasible direction vectors are defined and obtained in terms of first partial derivatives of the constraints. Necessary and sufficient conditions in terms of directional derivatives are derived and their equivalent with results in the literature. Sufficient conditions higher than second order can be also obtained and a Scheeffer type analysis for the solution of this problem can be found.