Calculus of Natural Logarithm: Integral Definition and Derivative Invariance

①This example dynamically illustrates the natural logarithm ln(x) defined as the integral of 1/x, and demonstrates that its derivative is independent of a constant scaling of the argument. ②Core mathematical concepts include the integral definition of the natural logarithm, the Fundamental Theorem of Calculus, the derivative of logarithmic functions, and the effect of scaling on derivatives. ③Construction highlights: using GeoGebra commands such as Point to create movable points, Integral to shade the area under the curve 1/x, Tangent and Slope to display tangent line slopes, and text objects (text1, text2, etc.) to label expressions and numerical relationships, with sliders a and b controlling integration limits and scaling factors. ④This resource is suitable for high school to undergraduate calculus courses, helping students grasp the connection between logarithms, integrals, and derivatives, and visually proving that d(ln x)/dx = d(ln(bx))/dx.