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Derivative of Arctangent Function

Derivative of Arctangent Function
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2016-10-21 05:48
FunctionTrigonometryCalculusDerivativeGeometric Construction
In collections BC Calculus
This example demonstrates the calculation and geometric interpretation of the derivative of the arctangent function f(x)=arctan(x). Key math concepts include derivatives, tangents, slopes, vectors, and trigonometry. Construction highlights involve using GeoGebra commands/objects like Point, Intersect, Circle, Tangent, Slope, Vector, and Textfield to build tangents, slope vectors, and function graphs. Suitable for high school calculus instruction and student exploratory learning.

GeoGebra Commands & Parameters

  1. Point(p) → A
  2. Intersect(i, yAxis) → B
  3. Intersect(f, i, (-1.2324462406838077, 0.39699494892935805)) → C
  4. Circle(C, 1) → c
  5. Tangent(C, f) → j
  6. Intersect(c, j, 1) → D
  7. Slope(j) → m
  8. Intersect(c, j) → F, G
  9. Circle(A, 0.9642295892450196) → d
  10. Tangent(A, p) → k
  11. Intersect(d, k) → H, I
  12. Vector(A, I) → u
  13. Vector(A, H) → v
  14. Textfield(n) → InputBox1

AI Drawing Prompt Reference

Draw the graph of y=arctan(x) and illustrate its derivative geometrically. First, plot a tangent line and find its intersection with the x-axis. Then, determine the tangent's direction by finding the intersection points between the tangent and a circle centered at the point of tangency. Next, draw a circle centered at the point of tangency and find its other intersection with the tangent line. Calculate the slope of the tangent and draw a perpendicular radius from the point of tangency. Label the derivative value.
The prompt above was compiled from the original drawing commands and is for reference only.