Differentiation from First Principles: Secant Slope Approaching Tangent Slope

This GeoGebra example dynamically illustrates the fundamental definition of the derivative from first principles, showing how the slope of a secant line approaches the slope of the tangent line as a second point moves toward the point of tangency. It visualizes a function f, a point A on the curve, and a secant line through A and another point B. As B slides along the curve toward A, the secant slope m converges to the tangent slope, making the limit concept of the derivative tangible. Core mathematical concepts include derivative, limit, secant and tangent lines, slope, and difference quotient. Construction details: the example uses GeoGebra commands Line to draw secant and tangent lines, Slope to display their slopes, and Tangent to generate the tangent line; key objects include coordinate points A and B, function f, parameter a, slope variables m and m_1, and a slider h controlling the horizontal distance between B and A for dynamic approximation. Suitable for high school and introductory college calculus students, this resource helps build an intuitive grasp of the derivative and serves as an effective classroom demonstration or self-study tool.