Learning Outcomes

  1. Understand the derivative of an inverse function.
  2. Calculate derivatives of the standard inverse trigonometric functions.

Textbook

  • Chapter 3.7   Derivatives of Inverse Functions  

Textbook Assignment

  • p. 306:    265, 267, 279-283 all, 287

WeBWork Assignment

  • Derivatives-Inverses

Exit problems of the session 

  1. Find the derivative of the following functions:

    (a).  y=\cos^{-1}(\sqrt{x})    (b).   y=x\tan^{-1}(3x)   (c).  y=(1+\sin^{-1}x)^3

 

Key Concepts

  • Differentiation formulas for the inverse trigonometric functions.

\dfrac{d}{dx}(\sin^{-1}x)=\dfrac{1}{\sqrt{1-x^2}}

\dfrac{d}{dx}(\cos^{-1}x)=\dfrac{-1}{\sqrt{1-x^2}}

\dfrac{d}{dx}(\tan^{-1}x)=\dfrac{1}{1+x^2}

\dfrac{d}{dx}(\cot^{-1}x)=\dfrac{-1}{1+x^2}

\dfrac{d}{dx}(\sec^{-1}x)=\dfrac{1}{|x|\sqrt{x^2-1}}     

\dfrac{d}{dx}(\csc^{-1}x)=\dfrac{-1}{|x|\sqrt{x^2-1}} 

 

Videos and Practice Problems of Selected Topics

  1. \rhd Derivatives of inverse functions from equation (5:03) Given f(x) =\dfrac{1}{2}x^3+3x-4 and let h be the inverse function of f.  Notice that f(-2)=-14.  Find h'(-14).
  2. \rhd Derivatives of inverse sine function (4:55) Deriving the formula of inverse sine function, \sin^{-1}x
  3. \rhd Derivatives of inverse cosine function (3:43) Deriving the formula of inverse cosine function, \cos^{-1}x.
  4. \rhd Derivatives of inverse tangent function (6:01) Deriving the formula of inverse tangent function, \tan^{-1}x.
  5. * Practice: Derivatives of inverse trigonometric functions. (4 problems)