The goal of this paper is the combined use of two theories, APOS and OSA, for the analysis of the university students' understanding on the graph of the function and its derivative. For this, we study the students' understanding to solve one graphing problem in relation to the first derivative and characterize their schemas in terms of levels (intra, inter and trans) of development of the schema for sketching ′ when given the graph . We present a multiple case study in which 14 students of the first course of Calculus in one university of Iran participated voluntarily. Results show that most of the students in our study had major problems in developing mental constructions and doing the practical work needed to solve the problem, particularly those mental constructions that have to be made to calculate the derivative at the critical points and to determine the speed of the variation of the inclination of the tangent lines to , which is why most of them have constructed a schema at the intra level of development of the schema for sketching ′ when given the graph . We finish with some final conclusions.

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APOS-ACE (Action, Process, Object, and Schema-Activities, Classroom discussion, and Exercises) is applied in this article to explore the teaching and learning of derivative by giving emphasis on its graphical understanding. For this purpose, a Genetic Decomposition is developed based on the outcomes of previous studies and on our personal teaching experiences. An ACE cycle is designed with the help of the Maple software and implemented on a group of freshmen Iranian students (experimental group). The outcomes of this implementation are evaluated by comparing the performance of the experimental group to the performance of another equivalent student group (control group), to which the same subject was taught in the traditional, lecture-based way. Our findings demonstrated students' who were in the experimental group shown a better understanding of the derivate compared to the control group. Therefore, such ACE cycle with Maple could be used more frequently for teaching calculus, especially derivative.

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