Math 2.0

Special functions Theorem

Derivative of cosine

Builds on Sine and cosinePower series are differentiable inside the radius of convergence

Statement

The cosine function is differentiable on ℝ\mathbb{R}, and cos⁡′(x)=−sin⁡x\cos'(x) = -\sin x for all real xx.

Remarks

Differentiate the power series term by term. The derivative of (−1)nx2n(2n)!\frac{(-1)^n x^{2n}}{(2n)!} is −(−1)n−1x2n−1(2n−1)!-\frac{(-1)^{n-1} x^{2n-1}}{(2n-1)!} for n≥1n \ge 1. So sin⁡′′=−sin⁡\sin'' = -\sin and cos⁡′′=−cos⁡\cos'' = -\cos. Both functions solve f′′+f=0f'' + f = 0. Mathlib states it as Real.hasDerivAt_cos.

Reading path

To understand this page from the ground up, read these 54 pages in order. The foundations come first.

  1. Limit point of a set definition
  2. Local maximum definition
  3. Sequence definition
  4. Limit of a function definition
  5. Local minimum definition
  6. Real sequence definition
  7. Subsequence definition
  8. Continuity at a point definition
  9. Derivative definition
  10. Bounded sequence definition
  11. Cauchy sequence definition
  12. Geometric sequence definition
  13. Limit of a sequence definition
  14. Monotone sequence definition
  15. Series definition
  16. Continuous function definition
  17. Fermat's theorem on stationary points statement
  18. Limit superior definition
  19. Cauchy sequences are bounded statement
  20. Sum of a finite geometric series statement
  21. Monotone convergence theorem for sequences statement
  22. Every sequence has a monotone subsequence statement
  23. Convergent sequence definition
  24. Pointwise convergence definition
  25. Squeeze theorem statement
  26. Partial sum definition
  27. Power series definition
  28. Bolzano–Weierstrass theorem statement
  29. The sequence r^n tends to zero for |r| < 1 example
  30. Radius of convergence definition
  31. Convergent series definition
  32. Uniform convergence definition
  33. Sine and cosine definition
  34. Boundedness theorem statement
  35. Cauchy criterion for sequences statement
  36. Uniform convergence on a set definition
  37. Absolute convergence definition
  38. A series of nonnegative terms converges iff its partial sums are bounded statement
  39. Series of functions definition
  40. Sum of a series definition
  41. Extreme value theorem statement
  42. Uniformly Cauchy sequence of functions definition
  43. Geometric series example
  44. Comparison test statement
  45. Rolle's theorem statement
  46. Root test statement
  47. Weierstrass M-test statement
  48. Mean value theorem statement
  49. Cauchy–Hadamard theorem statement
  50. Power series converge uniformly on smaller discs inside the radius of convergence statement
  51. Uniform convergence of derivatives allows term-by-term differentiation statement
  52. Term-by-term differentiation statement
  53. Power series are differentiable inside the radius of convergence statement
  54. Derivative of cosine statement

Where this leads

23 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Differentiation_of_trigonometric_functions, Derivative of the cosine function
About this page
Tag0V2DV29EGT
URIm2://0V2DV29EGT
Kindstatement, theorem
WritingW0 stub
MSC33B10, 26A24
StatusThis copy of the site has no build data. Run m2 status 0V2DV29EGT.
UsesSine and cosine, Power series are differentiable inside the radius of convergence

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