<p>Exploring necessary optimality conditions is a crucial aspect of optimization. Various approaches exist to determine these conditions, including using subdifferentials rather than gradients. Recently, the concept of convexificators has emerged as a generalization of the subdifferential. A significant extension of this concept is the directional convexificator, which we further develop into the directional semi-regular convexificator. In our study, we investigate several calculus properties related to this new notion. We specifically apply it to examine the mean value theorem and characterize Lipschitz continuity through the lens of the directional semi-regular convexificator. Finally, we demonstrate the Fritz–John necessary optimality condition using this innovative concept.</p>