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Calculus - Derivative of Arctan (Geometric Proof)

Calculus - Derivative of Arctan (Geometric Proof)
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2016-12-29 00:28
CalculusDerivativeGeometric Construction
This example demonstrates the geometric proof of the derivative of the arctan function using GeoGebra. It illustrates how geometric shapes and transformations can be used to derive the derivative formula for arctan(x). Key math concepts include derivatives, geometric proofs, trigonometry, and limits. Construction highlights involve using GeoGebra commands and objects like Intersect, Segment, Circle, Ray, Angle, CircleSector, Polygon, Midpoint, Rotate. Suitable for high school and early university students learning calculus and geometric proofs.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → O
  2. Intersect(c, f, 1) → A
  3. Segment(O, F) → l
  4. Circle((0, 0), l) → d
  5. Intersect(d, f, 1) → D
  6. Segment(O, A) → g
  7. Segment(O, D) → h
  8. Ray(O, A) → i
  9. Intersect(d, i, 1) → C
  10. Intersect(c, h, 1) → B
  11. Segment(A, C) → j
  12. Intersect(f, xAxis) → E
  13. Angle(A, O, B) → θ
  14. Segment(D, E) → k
  15. CircleSector(O, A, B) → e
  16. CircleSector(O, C, D) → p
  17. Polygon(O, A, D) → poly1, d_1, o, a_1
  18. Segment(A, O) → m
  19. Segment(O, D) → n
  20. Midpoint(m) → G
  21. Midpoint(n) → H
  22. Rotate(H, π / 5, O) → H'
  23. Segment(A, E) → s
  24. Segment(O, E) → f_1
  25. Segment(D, E) → g_1
  26. Angle(E, O, D) → α
  27. Angle(E, O, A) → β
  28. Segment(O, A) → h_1
  29. Segment(O, D) → i_1
  30. Angle(A, E, O) → γ

AI Drawing Prompt Reference

Draw a geometric proof diagram for the derivative of arctangent. Start with a circle centered at the origin, intersecting the x-axis and a slanted line through the origin. Create two rays from the origin meeting the circle and the line, forming two circular sectors. Connect points to form a triangle and several segments. Mark key angles. Take midpoints of two triangle sides, rotate one midpoint around the origin by a fixed angle. Finally, add auxiliary segments from circle points to axis intersections, and label multiple angles to illustrate the geometric relationships.
The prompt above was compiled from the original drawing commands and is for reference only.