Visualizing Differentiation – Exploring Polynomial Derivatives

① This example visualizes the differentiation of polynomials up to the fourth order, illustrating the relationship between the original function, its derivative, and geometric interpretations such as tangent lines and slopes. ② Core mathematical concepts include differentiation from first principles, derivatives of polynomial functions, slope of tangent lines, quadratic equations, higher‑order derivatives, and the connection between algebraic and geometric representations. ③ Construction details: GeoG2 uses commands like Tangent (to draw tangent lines), Slope (to compute line slopes), Line (to create lines), Point (to define points), Segment (to display segments), OrthogonalLine (to draw perpendicular lines), and Derivative (to compute symbolic derivatives). Objects such as movA, movB (sliders for polynomial coefficients), e, a, b, c, d, e_1 (coefficients), and text objects text2, text3, text8 are employed to dynamically adjust coefficients and display results. ④ Suitable audience: high‑school and early university students learning calculus, teachers preparing lessons on differentiation, and anyone interested in interactive visualizations of mathematical concepts. It serves as an effective classroom demonstration, online tutoring tool, or self‑study resource for exploring the geometric meaning of derivatives.