From a physical perspective, derivatives can be viewed as mathematical idealizations of the linear growth as expressed by the Lipschitz condition. On the other hand, non-linear local growth conditions have been also proposed in the literature. The manuscript investigates the general properties of the local generalizations of derivatives assuming the usual topology of the real line. The concept of a derivative is generalized in terms of the local growth condition of the primitive function. These derivatives are called modular derivatives. Furthermore, the conditions of existence of the modular derivatives are established. The conditions for the continuity of the generalized derivative are also demonstrated. Finally, a generalized Taylor–Lagrange property is proven. It is demonstrated that only the Lipschitz condition has the special property that the derivative function is non-trivially continuous so that the derivative does not vanish.

从物理角度看，导数可视为Lipschitz条件表示的线性增长的数学理想化。另一方面，文献中也提出了非线性局部生长条件。该手稿假设实线具有通常的拓扑结构，研究了导数的局部概括的一般属性。根据原始函数的局部增长条件来概括导数的概念。这些导数称为模块化导数。此外，确定了模块化导数的存在条件。还证明了广义导数连续的条件。最后，证明了广义泰勒-拉格朗日性质。