The Definition of the Derivative: Secant to Tangent

This interactive GeoGebra example visually demonstrates the fundamental definition of the derivative: as a moving point Q approaches a fixed point P along a function curve, the slope of the secant line PQ converges to the slope of the tangent line at P, which is the derivative. It illustrates key concepts such as limits, secant lines, tangent lines, slopes, and the geometric interpretation of derivatives. The construction uses GeoGebra commands including Point to define points P and Q on the function f, Derivative to generate the derivative function, Line and OrthogonalLine to create secant and tangent lines, Intersect for intersection points, Polygon for auxiliary triangles, Textfield and inputBox1 for interactive input, Slider to control Q's position, and Text objects to display numerical values. Suitable for high school and introductory college calculus students, this resource helps learners grasp the limit-based definition of derivatives and serves as an effective classroom demonstration tool.