Differential and derivative are core concepts in analytical mathematics and are closely related. To be brief, the derivative refers to the rate of change of a function near a point, while the differential describes the actual change in the value of the function under the action of the derivative. Together, they form the basis of calculus. Differential is an application of the concept of derivative, specifically the response to a small change in the function. The derivative is the slope of the local linear approximation, while the differential is the increment of the function value under this linear approximation. This paper mainly discusses the topic about derivative and differential. To illustrate them clearly, this paper will introduce step by step, from their concept to examples, including definition, properties and geometric meaning. Specifically, the article particularly emphasizes applications in sequence problems and approximate calculations. These applications are of great significance in mathematics, physics, engineering, and economics.
