The concept of the derivative originates from the idea of measuring how a function changes at an infinitesimally small scale. Formally, the derivative of a function at a point is defined as the limit of the average rate of change as the interval shrinks to zero. This definition captures the notion of an instantaneous rate of change and provides the foundation for differential calculus.
This limit-based formulation allows us to rigorously analyze slopes of tangent lines, rates of variation in physical systems, and dynamic behaviors in engineering and science.
