Derivatives of Inverse Trigonometric Functions

Derivatives of Inverse Trigonometric Functions:

  1. Inverse Sine Function (arcsin(x):

    • ddx[arcsin(x)]=11x2for 1<x<1
      • Derivative of arcsin(x) is positive for x in the open interval (-1, 1).
  2. Inverse Cosine Function (arccos(x):

    • ddx[arccos(x)]=11x2 for 1<x<1

                Derivative of arccos(x) is negative for x in the open interval (-1, 1).

     3. Inverse Tangent Function (arctan(x):

    • ddx[arctan(x)]=11+x2
    • The derivative of arctan(x) is always positive.

Derivatives of Other Inverse Trigonometric Functions:

  • Derivative of Inverse Cotangent Function (arccot(x):

    • ddx[arccot(x)]=11+x2
  • Derivative of Inverse Secant Function (arcsec(x):

    • ddx[arcsec(x)]=1xx21 for x>1
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  • Derivative of Inverse Cosecant Function (arccsc(x):

    • ddx[arccsc(x)]=1xx21
    • for x>1

Chain Rule with Inverse Trigonometric Functions:

Chain Rule for Inverse Trigonometric Functions:

  • When the inverse trigonometric function is nested within another function, the chain rule applies.
    • Example: f(x)=arcsin(2x2)
    • f(x)=4x14x4
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Applications of Inverse Trigonometric Functions in Differentiation:

  • Used in integration to solve integrals involving algebraic expressions and square roots.
  • Often utilized in physics, engineering, and mathematical modeling to describe inverse relationships in various phenomena.