Action-Process-Object-Schema (APOS) theory and tools resulting from dialogue with the Anthropological Theory of the Didactic (ATD) were used to analyse data from semi-structured interviews and teaching materials to study students' understanding of the relationship between tangent planes and the total differential. Results of the study show students' difficulties relating these ideas and suggest a refinement of the initial genetic decomposition. They also underline aspects of the teaching materials that need to be considered to promote those constructions and development of a complete praxeology for the total differential. This study exemplifies how the dialogue between a cognitive theory and one that focuses on institutional aspects of mathematics education, can provide tools to deeply analyse the teaching and learning of a mathematical topic.