Derivatives and Differentials of Multivariable Functions: A Theoretical and Applied Analysis

Abstract

This article presents a comprehensive theoretical and applied investigation of derivatives and differentials of multivariable functions within the framework of higher mathematics. The study examines partial derivatives of functions of two and n variables, develops the concepts of the total differential and the gradient, and establishes the key properties governing these operations, including linearity, the product and quotient rules, the chain rule, and the equality of mixed partial derivatives (Clairaut’s theorem). The theoretical exposition is supported by three worked examples covering polynomial, trigonometric, and logarithmic multivariable functions. The geometric interpretation of the total differential as the tangent-plane approximation is discussed alongside its practical applications in physics, chemistry, and engineering. The findings confirm that multivariable derivatives and differentials constitute a foundational tool for local behaviour analysis, extremum problems, and linear approximation in applied science and engineering contexts.