Explore Derivatives of a 5th-Order Polynomial Equation

① This interactive GeoGebra example lets students manipulate the coefficients of a 5th-order polynomial equation and observe how its derivative graphs change in real time. ② It demonstrates core calculus concepts: polynomial functions, first and higher-order derivatives, tangent and normal lines, extrema, inflection points, and the relationship between a function and its derivatives. ③ The construction uses GeoGebra commands such as Point, Tangent, OrthogonalLine, and Intersect to build the original function f, its first derivative f', second derivative f'', and higher-order derivatives (k', k'', k''', etc.). Sliders (a, b, c, d, e) control the coefficients, while tangent and normal lines are drawn at selected points, with labeled points (A, B, C, D, F, H, I, J, K, L) and text annotations. ④ It is suitable for high school or early college calculus courses, helping students visualize the meaning of derivatives and explore how higher-order derivatives affect curve shape.