Geometric Definition of the Derivative at a Point

This example demonstrates how the derivative of a function at a point is obtained geometrically: the slope of a secant line changes as the second point approaches the given point, and in the limit it converges to the slope of the tangent line. The key ideas are the pointwise definition of the derivative, the relationship between secant/tangent lines, and the use of similar triangles to interpret slope-related quantities as measurable geometric ratios. Construction relies on GeoGebra commands such as Tangent (to build the tangent line), Line (to create the secant line), Segment (to form the line segments used in the slope/similarity reasoning), SelectedIndex (to pick and update the active indices/points), and If/Element (to control conditional object selection and updates). Text labels (e.g., parameter and function annotations) are added to show the current values while using animation/reset controls (btnAnimate, btnReset, and show toggles) so learners can visually track the “secant-to-tangent” convergence. Suitable for calculus instruction and understanding derivatives and limits through geometric intuition.