Identifying the essential views and external contexts: the presentations and the arrangement of the interpretation of derivatives in textbooks for different student groups

High school calculus is a critical topic in mathematics education, and recent reviews have called for addressing diverse learning needs. While textbooks strongly shape students’ learning opportunities, previous studies have primarily examined representations or task characteristics, with less attention to how textbooks organize various derivative interpretations in pedagogically meaningful sequences. In response, this study analyzes three Taiwanese high school program textbooks oriented toward science, economics, and engineering. We distinguish essential views (fundamental conceptual meanings) from external contexts (representational and situational forms), grounded in Peirce’s semiotics and Duval’s theory of semiotic representation. Coding results and UpSet plots reveal patterns of use and co-occurrence of derivative interpretations across programs: the science-oriented textbook prioritizes the tangent slope view, whereas the vocational engineering textbook emphasizes the local rate of change, often with limited graphical connection. This study contributes an integrated lens for textbook research and offers implications for differentiated curriculum design in calculus education.