Derivative and 3D Graph Simulation in MATLAB: Mathematical Exploration and Visual Analysis

This study presents a MATLAB-based computational visualization system for the three-dimensional representation of multivariable functions and their partial derivatives. The proposed framework integrates symbolic differentiation, numerical evaluation, and 3D graphical rendering within a reproducible computational environment. The system, implemented in MATLAB (R2023b), uses the Symbolic Math Toolbox together with built-in visualization functions such as surf and quiver3. A benchmark function, z=x^2-y^2, is used to demonstrate the visualization of saddle surface geometry and corresponding derivative fields. Results show that the system effectively represents multivariable functions and their partial derivatives in a geometrically consistent manner. Unlike static visualization methods, the proposed approach enables integrated and dynamic exploration of mathematical structures. However, the study does not include empirical evaluation of learning outcomes; thus, educational implications are considered conceptual. Overall, the framework provides a reproducible and efficient tool for visualizing multivariable calculus concepts in a computational environment.