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微分除了直接用于近似计算以外,它的概念和运算在微积分课程中还有广泛的应用。如果能从多方面了解这些应用,就会进一步明确微分教学的目的,并可使有关内容的教学取得更好的效果。中学统编教材对近似计算方面的应用已作介绍,本文仅就微分与导数、不定积分、定积分等的关系,谈些个人见解,以供教学上参考。不当之处,还望大家指正。一、微分与导数的关系在中学教材中,微分的概念直接用导数来定义。可导函数y=f(x)的微分是 dy=f′(x)dx (1)其中dx作为记号,代表△x,并称为自变量的微分。如果x不作为自变量,而是任一变量t的可导函数x=φ(t),那么复合函数y=f[φ(t)]的微分就是
In addition to being used directly for approximate calculations, the concept of differentiation and its operations are widely used in calculus. If we can understand these applications from many aspects, we will further clarify the purpose of differential teaching and make the teaching of related content better. The application of textbooks for secondary school textbooks has been introduced to approximate calculations. This article only discusses the relationship between derivative and derivatives, indefinite integrals, definite integrals, etc. and discusses some personal opinions for reference in teaching. Improper, but also hope everyone correct me. First, the relationship between derivative and derivative In the middle school textbooks, the concept of differentiation is directly defined by the derivative. The derivative of the derivative function y = f(x) is dy = f’(x)dx (1) where dx serves as a token, representing Δx, and is called the derivative of the independent variable. If x is not an independent variable but a derivative function of any variable t is x = φ(t), then the derivative of the composite function y = f[φ(t)] is