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Mathematics Dictionary
Dr. K. G. Shih
Derivative of inverse trigonometric functions
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Answers


C01. Derivative of y = arcsin(x)

Formula
  • y = arcsin(x) and y' = 1/Sqr(1 - x^2)
Proof
  • Since y = arcsin(x), hence x = sin(y)
  • Take derivative on both sides with respective x
  • d/dx(x) = d/dx(sin(y))
  • 1 = (d/dy(sin(y)))*(dy/dx)
  • Hence dy/dx = 1/(cos(y))
  • Hence dy/dx = 1/Sqr(1 - sin(y)^2)
  • Hence dy/dx = 1/Sqr(1 - x^2)
Note
  • This is change trigonometric function to algebraic function

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C02. Derivative of y = arccos(x)

Formula
  • y = arccos(x) and y' = -1/Sqr(1 - x^2)
Proof
  • Since y = arccos(x), hence x = cos(y)
  • Take derivative on both sides with respective x
  • d/dx(x) = d/dx(cos(y))
  • 1 = (d/dy(cos(y)))*(dy/dx)
  • Hence dy/dx = -1/(sin(y))
  • Hence dy/dx = -1/Sqr(1 - cos(y)^2)
  • Hence dy/dx = -1/Sqr(1 - x^2)
Note
  • This is change trigonometric function to algebraic function

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G03. Derivative of y = arctan(x)

Formula
  • y = arctan(x) and y' = 1/(1+x^2)
Proof
  • Since y = arctan(x), hence x = tan(y)
  • Take derivative on both sides with respective x
  • d/dx(x) = d/dx(tan(y))
  • 1 = (d/dy(tan(y)))*(dy/dx)
  • Hence dy/dx = 1/(sec(y)^2)
  • Hence dy/dx = 1/(1 + tan(y)^2)
  • Hence dy/dx = 1/(1 + x^2)
Note
  • 1. This is change trigonometric function to algebraic function
  • 2. This formula is often used to find integral of y = 1/(a*x^2 + b*x + c)

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C04. Derivative of y = arccsc(x)


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C05. Derivative of y = arcsec(x)


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C06. Derivative of y = arccot(x)


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C07.


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C08. Answer

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Q09. Answer

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Q10. Answer

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