Why Calculus Begins With Functions: Understanding Change Through Mathematical Structure

Viewed through the lens of mathematics, the changes we observe in the world begin to fall neatly into place. Mathematics is a universal language—one that can be used anywhere in the universe. Among its many branches, calculus is the language that describes how the world changes. Seen from this perspective, calculus reveals the underlying principles of advanced technologies, natural phenomena, and even societal dynamics. Through differentiation, we grasp instantaneous change and motion; through integration, we understand how small increments accumulate to form the present state. In essence, integrating the past explains the present, and differentiating the present allows us to anticipate the future.

At its core, calculus is built on functions. This is because calculus is fundamentally a framework for measuring change and accumulation, both of which require a clear relationship between an input and ...