Connecting Theorems and Definitions: Students' Exploration of Function Derivatives

This study aims to analyze students' understanding of derivatives of functions through independent exploration, where students were asked to create examples of functions, calculate their derivatives using differentiation theorems, and prove the results using the definition of derivatives. The subjects of the study were 35 first-semester students from the mathematics education program. Data was collected through exploratory tasks, interviews, and observations. The results showed that all students were able to determine the derivatives of the functions they created, although proving the derivative using the definition was more challenging as it involved the concept of limits previously studied. Five students demonstrated a deeper understanding by creating more complex examples, such as root and rational functions, which required manipulation techniques and the application of the quotient rule for derivatives. Interviews revealed that proving the derivative with the definition provided a deeper understanding of the fundamental concepts of derivatives. This study suggests using varied exploratory tasks and reflection sessions to enhance students' conceptual understanding of derivatives.