Math 2.0

Special functions Proposition

Derivative of arcsine

Builds on ArcsineDerivative of an inverse functionDerivative of sinePythagorean trigonometric identity

Statement

The arcsine is differentiable on (−1,1)(-1, 1), and ddxarcsin⁡x=11−x2for −1<x<1.\frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}} \quad \text{for } -1 < x < 1.

Remarks

Let y=arcsin⁡xy = \arcsin x. The inverse function rule gives the derivative 1cos⁡y\frac{1}{\cos y}. By the Pythagorean identity, cos⁡y=1−x2\cos y = \sqrt{1 - x^2}, since cos⁡y>0\cos y > 0 for ∣y∣<π/2|y| < \pi/2. The arcsine is not differentiable at ±1\pm 1. Its graph has vertical tangents there. Similarly, ddxarccos⁡x=−11−x2\frac{d}{dx} \arccos x = -\frac{1}{\sqrt{1 - x^2}}. Mathlib states the main claim as Real.hasDerivAt_arcsin.

Reading path

To understand this page from the ground up, read these 69 pages in order. The foundations come first.

  1. Limit point of a set definition
  2. Local maximum definition
  3. Sequence definition
  4. Strictly monotone function definition
  5. Limit of a function definition
  6. Local minimum definition
  7. Real sequence definition
  8. Subsequence definition
  9. Continuity at a point definition
  10. Derivative definition
  11. One-sided limit definition
  12. Bounded sequence definition
  13. Cauchy sequence definition
  14. Geometric sequence definition
  15. Limit of a sequence definition
  16. Monotone sequence definition
  17. Series definition
  18. Continuous function definition
  19. Epsilon-delta definition of continuity statement
  20. Fermat's theorem on stationary points statement
  21. Limit superior definition
  22. Cauchy sequences are bounded statement
  23. Every sequence has a monotone subsequence statement
  24. Sum of a finite geometric series statement
  25. Monotone convergence theorem for sequences statement
  26. Convergent sequence definition
  27. Pointwise convergence definition
  28. Squeeze theorem statement
  29. Partial sum definition
  30. Power series definition
  31. Intermediate value theorem statement
  32. Composition of continuous functions is continuous statement
  33. One-sided limits of a monotone function exist statement
  34. Bolzano–Weierstrass theorem statement
  35. The sequence r^n tends to zero for |r| < 1 example
  36. Radius of convergence definition
  37. Convergent series definition
  38. Uniform convergence definition
  39. Sine and cosine definition
  40. Chain rule statement
  41. Boundedness theorem statement
  42. Continuity of the inverse of a strictly monotone continuous function statement
  43. Cauchy criterion for sequences statement
  44. Absolute convergence definition
  45. Uniform convergence on a set definition
  46. A series of nonnegative terms converges iff its partial sums are bounded statement
  47. Series of functions definition
  48. Sum of a series definition
  49. Extreme value theorem statement
  50. Derivative of an inverse function statement
  51. Uniformly Cauchy sequence of functions definition
  52. Geometric series example
  53. Comparison test statement
  54. Rolle's theorem statement
  55. Root test statement
  56. Weierstrass M-test statement
  57. Mean value theorem statement
  58. Cauchy–Hadamard theorem statement
  59. Power series converge uniformly on smaller discs inside the radius of convergence statement
  60. A function with zero derivative is constant statement
  61. Uniform convergence of derivatives allows term-by-term differentiation statement
  62. Term-by-term differentiation statement
  63. Power series are differentiable inside the radius of convergence statement
  64. Derivative of cosine statement
  65. Derivative of sine statement
  66. Pythagorean trigonometric identity statement
  67. pi object
  68. Arcsine definition
  69. Derivative of arcsine statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Differentiation_of_trigonometric_functions, Differentiating the inverse sine function
About this page
Tag0GA6G43EA9
URIm2://0GA6G43EA9
Kindstatement, proposition
WritingW0 stub
MSC33B10, 26A24
StatusThis copy of the site has no build data. Run m2 status 0GA6G43EA9.
UsesArcsine, Derivative of an inverse function, Derivative of sine, Pythagorean trigonometric identity

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.