Foundations of Calculus: Limits and Derivatives

Calculus is the mathematical language of change, but mastering it requires a crystal-clear understanding of its two fundamental pillars: limits and derivatives. This written course guides you step-by-step through these essential concepts, transforming abstract formulas into intuitive, practical knowledge. By reading through our structured explanations and working through guided mathematical exercises, you will develop the confidence to analyze functions, calculate rates of change, and solve foundational calculus problems with ease. What you'll learn: Understand the fundamental concept of a limit and how functions behave near specific points; Calculate limits using algebraic properties, factoring, rationalization, and standard limit theorems; Define continuity and identify points of discontinuity in various functions; Master the definition of the derivative as an instantaneous rate of change; Apply core differentiation rules, including the power, product, quotient, and chain rules; Explore modern applications of derivatives in optimization, physics, and basic machine learning algorithms. The journey begins with key terminology, defining what a limit is before moving on to continuity and the definition of a derivative. You will progress from basic algebraic manipulation to applying differentiation rules to real-world rate-of-change scenarios. This course is designed for absolute beginners, high school students, or anyone looking to refresh their mathematical foundations before diving into advanced calculus, physics, or data analysis. No prior calculus experience is required. Start reading today to unlock the power of calculus and elevate your mathematical problem-solving skills.