Calculus is a very fundamental subject for Mathematics education students (Mahir & N., 2009)(Funny & A., 2021)(Aniswita & A, 2023). According to (Dunham & W., 2005), the concepts in Calculus are very important, broad and amazing and are considered the greatest discovery of Modern mathematics (Kidron & I., 2014). Calculus can solve various problems in various fields such as science, engineering, medicine and economics (Latorre et al., 2007)(Alam & A., 2020)(Stevens & N., 2021). So that the Calculus course is important to study in various study programs that make the concept of Calculus a prerequisite, especially the Mathematics Education study program (Funny & A., 2021)(Hashemi et al., 2015)(Dunham & W., 2005)(Mahir & N., 2009)(Rasmussen et al., 2014)(Vajravelu et al., 2016). The concepts in Calculus are considered to support students' analytical abilities and bridge basic mathematics to higher-level mathematics (Dunham & W., 2005)(Mahir & N., 2009).

One of the important concepts in Calculus is the concept of derivatives (Varberg et al., 2016)(Hashemi et al., 2014)(Kidron & I., 2014)(Mkhatshwa & T., 2024). The concept of derivatives is the basis for understanding other Calculus concepts, such as the concept of Indefinite Integrals. So that mastery of the concept of derivatives is a must. In fact, this concept is still considered a difficult concept (White et al., 1996)(Aniswita & R., 2016)(Deswita et al., 2021). Quite a lot of research reveals the difficulties experienced by students in understanding the concept of derivatives. Among them, research conducted by (Orton & A., 1983) which was the first study to report students' difficulties in understanding the concept of derivatives. There is a significant misunderstanding regarding the definition of derivative and the graphical representation of derivatives. The same thing was also found by (García-García & Dolores-Flores, 2021) that students have difficulty interpreting derivative graphs. Added by (Hashemi et al., 2014) and (Denbel & G., 2015) that the difficulty is because students focus too much on the symbolic aspect rather than the graphic aspect and connecting the two. This is also reinforced by research (Ferrini-Mundy et al., 1994) who found students had difficulty connecting symbolic representation with graphical understanding. In addition, (Mkhatshwa & T., 2020) also revealed that students had difficulty using derivative rules in calculating derivatives.

This difficulty stems from students' lack of understanding, which is caused by their low level of conceptual knowledge (Mahir & N., 2009)(Hashemi et al., 2014). Conceptual knowledge can be simply defined as knowledge about a concept. According to (Kilpatrick et al., 2001), conceptual knowledge refers to an understanding of mathematical concepts, operations, and relationships. This is in line with (Schneider et al., 2010), who define conceptual knowledge as abstract understanding of principles and relationships between parts of knowledge within a specific domain. This definition is further reinforced by (Khashan & K.H., 2014), who describes conceptual knowledge as abstract knowledge that addresses the nature of mathematical principles and the relationships among those principles. (İşleyen et al., 2003) describe conceptual knowledge in mathematics as knowledge consisting of symbols and demonstrations. Similarly, (Baroody et al., 2007) define conceptual knowledge as knowledge of concepts and principles, as well as the relationships between them. Conceptual knowledge is rich in relationships and forms a network of interconnected knowledge (Hiebert et al., 1986). It can be interpreted as an understanding of the relationships among concepts, definitions, and mathematical rules, and the ability to explain them (Zuya, 2017).

A lack of conceptual knowledge can hinder students' ability to transfer and generalize knowledge (Hurrell, 2021). Both of these skills are crucial in learning Calculus. Therefore, it is important to understand students' conceptual understanding of Calculus concepts, particularly the concept of derivatives. This is in line with the opinions of (Zuya, 2017) and (Masduki et al., 2023) regarding the importance of identifying and analyzing how students understand conceptual knowledge as a basis for designing meaningful learning.

Research on students' conceptual understanding of derivatives is still very limited. Most focus on the definition, symbols, graphs, and applications of derivatives, while other important concepts remain unexplored. Therefore, it is necessary to fully uncover students' conceptual understanding of derivatives. It is hoped that this study can serve as a foundation for improving Calculus instruction, enabling students to better understand Calculus concepts, especially derivatives, and making learning more meaningful. As (Schoenfeld & H., 1995) stated, "We believe that developing conceptual understanding, not algebraic technique, should be the driving force, and we expect students to engage with mathematics."

This study employs a qualitative research approach. According to (Bogdan et al., 1992), qualitative research is a procedure that generates descriptive data in the form of spoken or written words, or observable behavior from the research subjects. This aligns with the research objective, which is to examine the conceptual understanding of Mathematics Education students at university on the topic of derivatives.

The research involved 62 students who took the Differential Calculus course. Students were given three questions designed to assess conceptual knowledge on the topic of derivatives. The questions were as follows:

Students’ answers were then categorized into four groups based on the accuracy of the concepts they applied in solving the problems. The categories are shown in Table 1.

To gain more detailed and in-depth insights into students' conceptual knowledge, the researchers conducted interviews with selected students. Four students were chosen to represent each category for every question, labeled with the question number and category code (e.g., 1K1, 1K2,

etc.). The collected data were analyzed using qualitative data analysis techniques based on (Miles et al., 2014), which consist of three stages: data reduction, data display, and conclusion drawing.

The results showed that most students struggled to solve the given problems. The distribution of answers is shown in Table 2. It shows that 65.1% of students did not understand the concept of derivatives. Only 15% were able to correctly answer the questions using appropriate concepts.

This research reveals students' conceptual understanding across three core concepts in derivatives: definition, theorems (rules), and implicit differentiation. Overall, students exhibited weak conceptual understanding. This supports by (Hashemi et al., 2014) findings that many students could compute derivatives but lacked a solid grasp of derivative concepts. (García-García & Dolores-Flores, 2021) also found that students struggled to connecting the concept of derivatives with the interpretation of derivative graphs.

The study shows that students have difficulty using the multiplication theorem to calculate the derivative of a function. This is in line with (Mkhatshwa & T., 2020) research which found that students still have difficulty using the derivative theorem to determine the derivative of a function. However, the definition and derivative of implicit functions are more difficult for students than applying derivative theorems. This reflects findings by (Puspita et al., 2023) who noted students often solve derivative problems based on memorized procedures rather than conceptual understanding. Students solve derivative problems from habit, not comprehension.

Students’ lack of understanding of limits, ratios, and proportions contributes to the difficulty in grasping the definition of derivatives (Byerley et al., 2012). Conceptual knowledge involves linking mathematical ideas—something many students fail to do. (Hashemi et al., 2014) also noted that students struggle to connect mathematical symbols with the quantities they represent. In terms of the product rule, students often misapplied it by multiplying individual derivatives rather than applying the correct formula. This indicates they could not distinguish between rules for addition and multiplication. Similarly, in implicit differentiation, students struggled due to their familiarity with only explicit functions. They focused on manipulating symbols rather than understanding the varying quantities those symbols represented (Hashemi et al., 2014).

Identifying these difficulties is crucial for designing more meaningful Calculus instruction tailored to students' needs. According to Vygotsky as cited in (Unknown Author, 2008), students often possess fragmented conceptual understanding that needs to be developed into structured knowledge. Understanding the needs of students helps teachers design (Masduki et al., 2023) appropriate and meaningful calculus learning. This research is still limited to three important topics and the subjects are still limited. Future researchers could develop other important topics and expand the subjects of research.

This study concludes that students’ conceptual understanding of derivatives is still low. Most students were unable to apply appropriate concepts to solve the problems. Specifically: For definition of derivatives, many students did not understand that a derivative represents the rate of change or the limit of the average rate of change. Many confused function values with derivative values. For product rule, many students mistakenly believed the derivative of a product is the multiplication of each of its derivatives (f(x).g(x))′ = f′(x).g′(x). For implicit differentiation, most students did not understand the structure of implicit functions or how to differentiate them. They often ignored the variable , assuming all functions should be expressed in terms of only.

The researchers would like to thank the Universitas Islam Negeri Bukittinggi for the support and facilities provided.