Math 2.0

Special functions Proposition

Derivatives of the hyperbolic functions

Builds on Hyperbolic sineHyperbolic cosineHyperbolic tangentThe exponential function is its own derivativeQuotient rulecosh^2 - sinh^2 = 1

Statement

The functions sinh⁡\sinh, cosh⁡\cosh and tanh⁡\tanh are differentiable on ℝ\mathbb{R}, and for every real xx sinh⁡′(x)=cosh⁡x,cosh⁡′(x)=sinh⁡x,tanh⁡′(x)=1cosh⁡2x=1−tanh⁡2x.\sinh'(x) = \cosh x, \qquad \cosh'(x) = \sinh x, \qquad \tanh'(x) = \frac{1}{\cosh^2 x} = 1 - \tanh^2 x.

Remarks

The first two follow from exp⁡′=exp⁡\exp' = \exp and (e−x)′=−e−x(e^{-x})' = -e^{-x}. The third follows from the quotient rule and the identity cosh⁡2−sinh⁡2=1\cosh^2 - \sinh^2 = 1. So sinh⁡′′=sinh⁡\sinh'' = \sinh and cosh⁡′′=cosh⁡\cosh'' = \cosh. Mathlib states the first one as Real.hasDerivAt_sinh.

Reading path

To understand this page from the ground up, read these 43 pages in order. The foundations come first.

  1. Limit point of a set definition
  2. Sequence definition
  3. Limit of a function definition
  4. Real sequence definition
  5. Continuity at a point definition
  6. Derivative definition
  7. Bounded sequence definition
  8. Geometric sequence definition
  9. Limit of a sequence definition
  10. Monotone sequence definition
  11. Series definition
  12. Differentiable function definition
  13. Sum of a finite geometric series statement
  14. Monotone convergence theorem for sequences statement
  15. Convergent sequence definition
  16. Squeeze theorem statement
  17. Cauchy product definition
  18. Partial sum definition
  19. Power series definition
  20. Differentiable functions are continuous statement
  21. The sequence r^n tends to zero for |r| < 1 example
  22. Radius of convergence definition
  23. Convergent series definition
  24. Product rule statement
  25. Reciprocal rule statement
  26. A series of nonnegative terms converges iff its partial sums are bounded statement
  27. Absolute convergence definition
  28. Sum of a series definition
  29. Quotient rule statement
  30. Comparison test statement
  31. Geometric series example
  32. Cauchy product of absolutely convergent series statement
  33. Ratio test statement
  34. Ratio formula for the radius of convergence statement
  35. The exponential series has infinite radius of convergence statement
  36. Exponential function definition
  37. Hyperbolic cosine definition
  38. Hyperbolic sine definition
  39. Exponential of a sum statement
  40. Hyperbolic tangent definition
  41. cosh^2 - sinh^2 = 1 statement
  42. The exponential function is its own derivative statement
  43. Derivatives of the hyperbolic functions statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Hyperbolic_functions, Derivatives
About this page
Tag07T2TKGZ4J
URIm2://07T2TKGZ4J
Kindstatement, proposition
WritingW0 stub
MSC33B10, 26A24
StatusThis copy of the site has no build data. Run m2 status 07T2TKGZ4J.
UsesHyperbolic sine, Hyperbolic cosine, Hyperbolic tangent, The exponential function is its own derivative, Quotient rule, cosh^2 - sinh^2 = 1

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