Derivatives are an important tool for single-objective optimization. In fact, it is commonly accepted that derivative-based methods present a better performance than derivative-free optimization approaches. In this work, we will show that the same does not always apply to multiobjective derivative-based optimization, when the goal is to compute an approximation to the complete Pareto front of a given problem. The competitiveness of direct multisearch (DMS), a robust and efficient derivative-free optimization algorithm, will be stated for derivative-based multiobjective optimization (MOO) problems, by comparison with MOSQP, a state-of-art derivative-based MOO solver. We will then assess the potential enrichment of adding first-order information to the DMS framework. Derivatives will be used to prune the positive spanning sets considered at the poll step of the algorithm. The role of ascent directions, that conform to the geometry of the nearby feasible region, will then be highlighted.