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Derivatives of Inverse Functions Demo

Derivatives of Inverse Functions Demo
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2016-10-24 19:08
FunctionCalculusDerivativeTransformationGraphing
① This example demonstrates the geometric relationship that the slopes of tangent lines to a function and its inverse at corresponding points are reciprocals. ② Core concepts: inverse functions, derivative definition, tangent slope, symmetry of graphs about y=x. ③ Construction highlights: Mirror reflects f across y=x to obtain g; Tangent draws tangents, Slope computes slopes, Vector shows direction, Intersect finds intersection points, InputBox and button2 enable interactive parameter control, while Circle, Point, Textfield provide annotations. ④ Suitable for high‑school calculus electives, introductory college calculus, and self‑study visualization.

GeoGebra Commands & Parameters

  1. Mirror(f, g) → f_1
  2. Textfield(f) → InputBox1
  3. Mirror(h, g) → h_1
  4. Textfield(a) → InputBox2
  5. Textfield(b) → InputBox3
  6. Mirror(f, g) → f_2
  7. Point(f) → A
  8. Tangent(A, f) → i
  9. Slope(i) → m
  10. Circle(A, 2) → k
  11. Intersect(k, i, 1) → B
  12. Intersect(k, i, 2) → C
  13. Vector(A, C) → u
  14. Vector(A, B) → v
  15. Intersect(j, f_2) → D, E
  16. Tangent(D, f_2) → l
  17. Circle(D, 2) → p
  18. Intersect(p, l, 2) → F
  19. Intersect(p, l, 1) → G
  20. Vector(D, F) → w
  21. Vector(D, G) → n
  22. Slope(l) → m_1

AI Drawing Prompt Reference

Draw a function curve and its inverse reflected across the line y=x. Place a point on the function, construct its tangent line and a circle of radius 2 centered at that point. Mark the two intersections of the circle with the tangent, and draw vectors from the point to these intersections. Repeat the same construction at the corresponding point on the inverse curve. Display the slopes of both tangent lines. Include three input boxes to adjust the function expression and parameters a and b, allowing interactive exploration of the reciprocal slope relationship.
The prompt above was compiled from the original drawing commands and is for reference only.