The Total Derivative of Multivariate Functions and Its Applications
School of Science, Guangdong University of Petrochemical Technology, Maoming 525000, China
| Abstract: | In this paper, we consider the concept of total derivatives of functions and its relationship with continuity, differentiability, and other concepts. The results indicate that: first, the existence of the total derivative of a bivariate function is a sufficient but not necessary condition for the function's continuity; second, in general, there is no mutually implicative relationship between the existence of the total derivative and the existence of partial derivatives for bivariate functions. However, if the total derivative of a bivariate function equals zero, then the partial derivatives with respect to both independent variables exist and are also zero; third, generally, there is no implicative relationship between the existence of the total derivative and differentiability for bivariate functions. Nevertheless, the total derivative being equal to zero is equivalent to the function being differentiable at that point and the total differential also being equal to zero; fourth, the existence of the total derivative of a bivariate function is a sufficient but not necessary condition for the existence of the directional derivative along any direction at that point. Finally, we presents an application of total derivatives——the mean value theorem, enriching the theory of differential calculus of multivariate functions. |
| Keywords: | Total Derivative; Partial Derivative; Continuity; Differentiability |
| DOI: | 10.57237/j.wjms.2025.02.003 |
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Received: 1 September 2025
Accepted: 23 October 2025
Published Online: 12 November 2025
Funding
| 1. | 广东石油化工学院自然科学研究项目 (2019rc101) |
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