莱布尼茨公式的图示方法及推广

Graphical Method for the Leibniz Formula and its Generalization

为解释高阶导数莱布尼茨公式和二项式定理形式上的相似性,提出了一种图形化的分析和证明思路.通过引入指数升幂算符和导数升阶算符将公式中的不同数学概念分离出来,进而在"树图"框架内对不同公式进行统一分析.相比于传统的"猜想-证明"方法,这里采用的方法具有简便、直观的优点,有利于加深学生对莱布尼茨公式的理解,并可直接导出多个函数乘积的高阶导数公式.

To explain the phenomenal similarity between the Leibniz formula for higher-order derivatives and the binomial theorem,agraphical methodology for analyzing and proving is proposed.By introducing promoting operators of exponent and derivative order,different mathematical concepts are isolated from original formulas,allowing uniform analysis under a same＂tree graph＂framework.Compared with traditional＂hypothesis-proof＂procedure,method adopted here is handy and straightforward,beneficial to student＇s comprehension of Leibniz formula,and can directly lead to the higher-order derivative formula for products of multiple functions.