Differentiability and Applications of Differentiation

Calculus is the mathematical language of change, but mastering its core principles requires a deep, intuitive understanding of how functions behave. This text-based course guides you from the fundamental definitions of limits and continuity to the practical applications of derivatives in real-world scenarios. By reading and working through the structured explanations and step-by-step mathematical proofs in this course, you will build a rock-solid foundation in single-variable calculus. You will transition from memorizing formulas to genuinely understanding the geometry and algebra behind differentiability, enabling you to tackle advanced engineering, physics, and data science problems. In this course, you will: Understand the formal definitions of limits, continuity, and differentiability from first principles; Apply key theorems, including Rolle's Theorem and the Mean Value Theorem, to analyze function behavior; Calculate derivatives of algebraic, trigonometric, exponential, and logarithmic functions step-by-step; Solve optimization problems by finding local and absolute extrema in various practical contexts; Analyze curves and sketch graphs using first and second derivative tests; Explore modern applications of differentiation, such as rate of change models and basic gradient concepts used in optimization algorithms. The course begins with foundational terminology, establishing a clear mathematical definition of limits and continuity before moving into the mechanics of finding derivatives. You will then explore rigorous applications, focusing on optimization, curve sketching, and rate of change problems through detailed written examples. This course is designed for beginners, college students starting engineering or science programs, and anyone looking to refresh their calculus fundamentals. No advanced mathematics background is required. Start reading today to unlock a deeper understanding of calculus and master the mechanics of change.