Calculating the Derivative at a Specific Point

Miacademy & MiaPrep Learning Channel

This educational video introduces the concept of the derivative at a specific point, bridging the gap between average rates of change and instantaneous rates of change. It begins by connecting the slope of a tangent line to the limit definition of a derivative, deriving the standard formula using 'h' to represent the distance between x-values. The narrator explains the notation $f'(a)$ and demonstrates how to calculate it algebraically through direct substitution and simplification limits. The video covers several key mathematical themes, including the definition of a derivative, evaluating limits, expanding binomials, and rationalizing numerators. It provides step-by-step worked examples ranging from polynomial functions to rational functions involving square roots. A significant portion of the video is dedicated to a real-world application problem involving projectile motion, where students learn to interpret the derivative as velocity (instantaneous rate of change) in the context of a rocket launch. For educators, this video serves as an excellent core lesson for Precalculus or Calculus I units on derivatives. It offers built-in pause points for student practice, addresses algebraic complexities like expanding powers and handling negative signs, and explicitly connects abstract mathematical calculations to physical interpretations like speed and direction. The visual graphs accompanying the algebraic work help students verify their calculations against geometric intuition.