Sketch Derivatives - 1: Visualizing f, f', and f'' with Interactive Input

This GeoGebra example demonstrates the graphical relationship between a function f, its first derivative f', and its second derivative f'' through an interactive input box. The core mathematical concept is the geometric interpretation of derivatives as slopes of tangent lines, and how the sign and zeros of derivatives correspond to monotonicity and inflection points of the original function. The construction uses the Textfield command to create an input box for user-defined functions, the Derivative command to automatically compute f' and f'', and the Intersect command to find key intersection points (A, B, P, C) between derivative curves and axes or other objects. Additional elements like an ellipse and centroid enrich the visual context. This resource is suitable for high school or introductory college calculus students, as well as teachers for classroom demonstrations, enabling learners to explore how changes in the function affect its derivative graphs and to build intuition about differentiation.