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Derivative from First Principles Demo

Derivative from First Principles Demo
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2020-03-16 18:20
FunctionCoordinate GeometryDerivativeGeometric Construction
This example demonstrates the geometric meaning of a derivative at a point, showing how the slope of a tangent line relates to the function. Key math concepts include derivative definition, limit concept, function graph, and tangent slope. GeoGebra commands used are Sequence, OrthogonalLine, Intersect, Segment, Angle, Point, If, and Midpoint. Suitable for high school students and above learning introductory calculus or understanding the relationship between derivatives and function graphs.

GeoGebra Commands & Parameters

  1. Sequence(u - x(A), u, b, q, -0.1) → List
  2. OrthogonalLine(B, xAxis) → i
  3. Intersect(i, xAxis) → C
  4. OrthogonalLine(A, xAxis) → j
  5. Intersect(j, xAxis) → D
  6. OrthogonalLine(B, yAxis) → k
  7. Intersect(k, yAxis) → F
  8. OrthogonalLine(A, yAxis) → l
  9. Intersect(l, yAxis) → E
  10. Intersect(i, l) → G
  11. Segment(B, C) → r
  12. Segment(A, D) → s
  13. Segment(G, E) → t
  14. Segment(B, F) → d
  15. Angle(G, A, B) → α
  16. Segment(A, G) → m_1
  17. Segment(B, G) → n
  18. Point(xAxis) → H
  19. Point(xAxis) → I
  20. Segment(H, I) → f_1
  21. Point(yAxis) → J
  22. Point(yAxis) → K
  23. Segment(J, K) → g_1
  24. Segment(E, A) → h_1
  25. Sequence(f(z) - y(A), z, b, q, -0.1) → L_1
  26. If(q ≤ 3, Element[L_1, 1] + "", "") → C3
  27. If(q ≤ 2.9, Element[L_1, 2] + "", "") → C4
  28. If(q ≤ 2.8, Element[L_1, 3] + "", "") → C5
  29. If(q ≤ 2.7, Element[L_1, 4] + "", "") → C6
  30. If(q ≤ 2.6, Element[L_1, 5] + "", "") → C7
  31. If(q ≤ 2.5, Element[L_1, 6] + "", "") → C8
  32. If(q ≤ 2.4, Element[L_1, 7] + "", "") → C9
  33. If(q ≤ 2.3, Element[L_1, 8] + "", "") → C10
  34. If(q ≤ 2.2, Element[L_1, 9] + "", "") → C11
  35. If(q ≤ 2.1, Element[L_1, 10] + "", "") → C12
  36. If(q ≤ 2, Element[L_1, 11] + "", "") → C13
  37. If(q ≤ 1.9, Element[L_1, 12] + "", "") → C14
  38. If(q ≤ 1.8, Element[L_1, 13] + "", "") → C15
  39. If(q ≤ 1.7, Element[L_1, 14] + "", "") → C16
  40. If(q ≤ 1.6, Element[L_1, 15] + "", "") → C17
  41. If(q ≤ 1.5, Element[L_1, 16] + "", "") → C18
  42. If(q ≤ 1.4, Element[L_1, 17] + "", "") → C19
  43. If(q ≤ 1.3, Element[L_1, 18] + "", "") → C20
  44. If(q ≤ 1.2, Element[L_1, 19] + "", "") → C21
  45. If(q ≤ 1.1, Element[L_1, 20] + "", "") → C22
  46. If(q ≤ 3, Element[List, 1]) → H3
  47. If(q ≤ 2.9, Element[List, 2]) → H4
  48. If(q ≤ 2.8, Element[List, 3]) → H5
  49. If(q ≤ 2.7, Element[List, 4]) → H6
  50. If(q ≤ 2.6, Element[List, 5]) → H7
  51. If(q ≤ 2.5, Element[List, 6]) → H8
  52. If(q ≤ 2.4, Element[List, 7]) → H9
  53. If(q ≤ 2.3, Element[List, 8]) → H10
  54. If(q ≤ 2.2, Element[List, 9]) → H11
  55. If(q ≤ 2.1, Element[List, 10]) → H12
  56. If(q ≤ 2, Element[List, 11]) → H13
  57. If(q ≤ 1.9, Element[List, 12]) → H14
  58. If(q ≤ 1.8, Element[List, 13]) → H15
  59. If(q ≤ 1.7, Element[List, 14]) → H16
  60. If(q ≤ 1.6, Element[List, 15]) → H17
  61. If(q ≤ 1.5, Element[List, 16]) → H18
  62. If(q ≤ 1.4, Element[List, 17]) → H19
  63. If(q ≤ 1.3, Element[List, 18]) → H20
  64. If(q ≤ 1.2, Element[List, 19]) → H21
  65. If(q ≤ 1.1, Element[List, 20]) → H22
  66. If(q ≤ 3, Element[L_1, 1]) → I3
  67. If(q ≤ 2.9, Element[L_1, 2]) → I4
  68. If(q ≤ 2.8, Element[L_1, 3]) → I5
  69. If(q ≤ 2.7, Element[L_1, 4]) → I6
  70. If(q ≤ 2.6, Element[L_1, 5]) → I7
  71. If(q ≤ 2.5, Element[L_1, 6]) → I8
  72. If(q ≤ 2.4, Element[L_1, 7]) → I9
  73. If(q ≤ 2.3, Element[L_1, 8]) → I10
  74. If(q ≤ 2.2, Element[L_1, 9]) → I11
  75. If(q ≤ 2.1, Element[L_1, 10]) → I12
  76. If(q ≤ 2, Element[L_1, 11]) → I13
  77. If(q ≤ 1.9, Element[L_1, 12]) → I14
  78. If(q ≤ 1.8, Element[L_1, 13]) → I15
  79. If(q ≤ 1.7, Element[L_1, 14]) → I16
  80. If(q ≤ 1.6, Element[L_1, 15]) → I17
  81. If(q ≤ 1.5, Element[L_1, 16]) → I18
  82. If(q ≤ 1.4, Element[L_1, 17]) → I19
  83. If(q ≤ 1.3, Element[L_1, 18]) → I20
  84. If(q ≤ 1.2, Element[L_1, 19]) → I21
  85. If(q ≤ 1.1, Element[L_1, 20]) → I22
  86. If(q ≤ 2.3, I10 / H10 + "", "") → E10
  87. If(q ≤ 1.9, Element[List, 12] + "", "") → D14
  88. If(q ≤ 3, Element[List, 1] + "", "") → D3
  89. If(q ≤ 2.9, Element[List, 2] + "", "") → D4
  90. If(q ≤ 2.8, Element[List, 3] + "", "") → D5
  91. If(q ≤ 2.7, Element[List, 4] + "", "") → D6
  92. If(q ≤ 2.6, Element[List, 5] + "", "") → D7
  93. If(q ≤ 2.5, Element[List, 6] + "", "") → D8
  94. If(q ≤ 2.4, Element[List, 7] + "", "") → D9
  95. If(q ≤ 2.3, Element[List, 8] + "", "") → D10
  96. If(q ≤ 2.2, Element[List, 9] + "", "") → D11
  97. If(q ≤ 2.1, Element[List, 10] + "", "") → D12
  98. If(q ≤ 2, Element[List, 11] + "", "") → D13
  99. If(q ≤ 1.8, Element[List, 13] + "", "") → D15
  100. If(q ≤ 1.7, Element[List, 14] + "", "") → D16
  101. If(q ≤ 1.6, Element[List, 15] + "", "") → D17
  102. If(q ≤ 1.5, Element[List, 16] + "", "") → D18
  103. If(q ≤ 1.4, Element[List, 17] + "", "") → D19
  104. If(q ≤ 1.3, Element[List, 18] + "", "") → D20
  105. If(q ≤ 1.2, Element[List, 19] + "", "") → D21
  106. If(q ≤ 1.1, Element[List, 20] + "", "") → D22
  107. If(q ≤ 3, I3 / H3 + "", "") → E3
  108. If(q ≤ 2.9, I4 / H4 + "", "") → E4
  109. If(q ≤ 2.8, I5 / H5 + "", "") → E5
  110. If(q ≤ 2.7, I6 / H6 + "", "") → E6
  111. If(q ≤ 2.6, I7 / H7 + "", "") → E7
  112. If(q ≤ 2.5, I8 / H8 + "", "") → E8
  113. If(q ≤ 2.4, I9 / H9 + "", "") → E9
  114. If(q ≤ 2.2, I11 / H11 + "", "") → E11
  115. If(q ≤ 2.1, I12 / H12 + "", "") → E12
  116. If(q ≤ 2, I13 / H13 + "", "") → E13
  117. If(q ≤ 1.9, I14 / H14 + "", "") → E14
  118. If(q ≤ 1.8, I15 / H15 + "", "") → E15
  119. If(q ≤ 1.7, I16 / H16 + "", "") → E16
  120. If(q ≤ 1.6, I17 / H17 + "", "") → E17
  121. If(q ≤ 1.5, I18 / H18 + "", "") → E18
  122. If(q ≤ 1.4, I19 / H19 + "", "") → E19
  123. If(q ≤ 1.3, I20 / H20 + "", "") → E20
  124. If(q ≤ 1.2, I21 / H21 + "", "") → E21
  125. If(q ≤ 1.1, I22 / H22 + "", "") → E22
  126. If(q ≤ 1, Element[L_1, 21]) → I23
  127. If(q ≤ 1, Element[List, 21]) → H23
  128. If(q ≤ 1, Element[L_1, 21] + "", "") → C23
  129. If(q ≤ 1, Element[List, 21] + "", "") → D23
  130. If(q ≤ 1, I23 / H23 + "", "") → E23
  131. Midpoint(C, D) → L
  132. Midpoint(F, E) → N
  133. Intersect(k, i_1) → O
  134. Intersect(l, i_1) → P
  135. Midpoint(O, P) → R
  136. Point(f) → T

AI Drawing Prompt Reference

Draw a function graph and demonstrate the geometric meaning of its derivative. Select a point on the function graph, draw the tangent line at that point, and display its slope. Also, use dynamically changing points to illustrate the derivative definition and limit concept, including parallel and perpendicular line segments to the x-axis, and their angle relationships.
The prompt above was compiled from the original drawing commands and is for reference only.