We establish two types of estimates for generalized derivatives of set-valued mappings which carry the essence of two basic patterns observed throughout the pile of calculus rules. These estimates also illustrate the role of the essential assumptions that accompany these two patters, namely calmness on the one hand and (fuzzy) inner calmness* on the other. Afterwards, we study the relationship between and sufficient conditions for the various notions of (inner) calmness. The aforementioned estimates are applied in order to recover several prominent calculus rules for tangents and normals as well as generalized derivatives of marginal functions and compositions as well as Cartesian products of set-valued mappings under mild conditions. We believe that our enhanced approach puts the overall generalized calculus into some other light. Some applications of our findings are presented which exemplary address necessary optimality conditions for minimax optimization problems as well as the calculus related to the recently introduced semismoothness* property.

我们为集值映射的广义导数建立了两种类型的估计，它们携带了在整个微积分规则堆中观察到的两种基本模式的本质。这些估计还说明了伴随这两种模式的基本假设的作用，即一方面是平静，另一方面是（模糊的）内心平静*在另一。然后，我们研究（内在）平静的各种概念之间的关系和充分条件。应用上述估计是为了在温和条件下恢复切线和法线以及边际函数和组合的广义导数以及集合值映射的笛卡尔积的几个突出的微积分规则。我们相信我们的增强方法将整体广义微积分置于其他方面。展示了我们发现的一些应用，这些应用示例性地解决了极小极大优化问题的必要优化条件以及与最近引入的半平滑*属性相关的微积分。