Mathematical modeling competency is essential for engineering students, yet significant cognitive obstacles impede their ability to apply theoretical concepts like derivatives to real-world optimization problems. This study investigates the cognitive processes and obstacles encountered by Industrial Engineering and Management students when solving applied derivative problems, utilizing the Mathematical Modeling Cycle (MMC) and Duval’s theory of semiotic registers as analytical frameworks. A qualitative case study design was employed, analyzing students’ written exam responses to an applied optimization task involving tour organization with variable pricing structures. Three representative cases were examined in detail, revealing distinct patterns of cognitive engagement. Results identified specific cognitive obstacles including misunderstanding of variables and domains, weak connections between mathematical and economic contexts, difficulties in graphical representation of constraints, and deficits in validation and critical thinking. While students demonstrated procedural fluency in symbolic manipulation and mathematical work, they struggled to coordinate between different semiotic registers (verbal, algebraic, graphical, and contextual) and failed to complete the full modeling cycle, particularly in the crucial validation stages. These findings suggest that cognitive obstacles stem from representational gaps rather than general learning difficulties, indicating the need for targeted pedagogical interventions that explicitly address transitions between semiotic registers and emphasize the iterative nature of mathematical modeling in engineering contexts.