The Action-Process-Object-Schema (APOS) theory is applied to study student understanding of implicit differentiation in the context of functions of one variable. The APOS notions of Schema and schema development in terms of the intra-, inter-, and trans-triad are used to analyze semi-structured interviews with 25 students who had just finished taking a single-variable calculus course. Results suggest that the notions of chain rule and implicit function play a key role in the possibility of attaining implicit differentiation Schema coherence. For this, students need to construct at least a Process conception of implicit function and a chain rule Schema with coherence given by function composition. Students also need to construct relations between implicit function and each one of three components of the implicit differentiation Schema: explicit function, derivative, and differentiation rules. The study shows that students taking an introductory calculus course can be expected to have difficulty understanding the main ideas of implicit differentiation unless special activities are designed to help them make the necessary connections between components of the implicit differentiation Schema. The study suggests the need to further investigate the implementation of activities that foster the constructions proposed, in textbooks and classrooms.

中文翻译：

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基于APOS理论的学生对内隐微分的理解

应用动作-过程-对象-模式（APOS）理论来研究学生在一个变量的函数背景下对隐性微分的理解。模式的 APOS 概念和模式发展在内部、内部和跨三元组方面用于分析对 25 名刚刚完成单变量微积分课程的学生的半结构化访谈。结果表明，链式法则和内隐函数的概念在获得内隐微分模式一致性的可能性中起着关键作用。为此，学生至少需要构建一个隐函数的过程概念和一个由函数组合给出的连贯性的链式规则模式。学生还需要构建隐函数与隐微分模式的三个组成部分中的每一个之间的关系：显式函数、导数和微分规则。该研究表明，参加微积分导论课程​​的学生可能难以理解内隐微分的主要思想，除非设计特殊活动来帮助他们在内隐微分模式的组成部分之间建立必要的联系。该研究表明，有必要进一步调查在教科书和课堂中促进所提议建筑活动的实施情况。研究表明，参加微积分导论课程​​的学生可能难以理解内隐微分的主要思想，除非设计特殊活动来帮助他们在内隐微分模式的组成部分之间建立必要的联系。该研究表明，有必要进一步调查在教科书和课堂中促进所提议建筑活动的实施情况。研究表明，参加微积分导论课程​​的学生可能难以理解内隐微分的主要思想，除非设计特殊活动来帮助他们在内隐微分模式的组成部分之间建立必要的联系。该研究表明，有必要进一步调查在教科书和课堂中促进所提议建筑活动的实施情况。