Differentiability of Functions for Beginners

Have you ever wondered how to mathematically calculate the exact rate of change at a single point on a curve? Understanding differentiability is the key to unlocking advanced calculus, physics, and modern machine learning algorithms. This text-based course guides you from basic geometric intuition to formal mathematical proofs of differentiability. You will transition from a basic understanding of limits to confidently analyzing the smoothness of various mathematical models. By reading through our structured explanations, you will learn how to identify where functions fail to be differentiable and how to apply derivative formulas to real-world rate-of-change problems. What you'll learn: - Understand the formal definition of a derivative using limits and rates of change - Identify points of non-differentiability such as corners, cusps, and discontinuities - Apply fundamental rules of differentiation including the chain rule and product rule - Analyze the relationship between continuity and differentiability in real-valued functions - Practice evaluating derivatives for polynomial, trigonometric, and exponential functions - Explore modern applications of derivatives in optimization and gradient-based algorithms This course begins with foundational concepts, defining the limit of a function and the geometric meaning of a tangent line. You will then progress through the algebraic mechanics of finding derivatives, exploring continuity, and analyzing piecewise functions for differentiability. This course is designed for beginners, high school students preparing for calculus, or university students looking for a clear, written refresher on mathematical analysis. No advanced calculus background is required. Start reading today to build a strong mathematical foundation in calculus.