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3.4 Differentiating Inverse Trigonometric Functions

Part of Differentiation: Composite, Implicit, and Inverse Functions in AP® Calculus AB. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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The inverse trigonometric functions also have standard derivative formulas that are usually developed using implicit differentiation. The most important are
ddx(arcsin⁡x)=11−x2,ddx(arccos⁡x)=−11−x2,ddx(arctan⁡x)=11+x2\frac{d}{dx}(\arcsin x)=\frac{1}{\sqrt{1-x^2}}, \frac{d}{dx}(\arccos x)=\frac{-1}{\sqrt{1-x^2}}, \frac{d}{dx}(\arctan x)=\frac{1}{1+x^2}
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These functions are useful because they appear whenever we solve trig equations for angles, and their derivatives come up often in applications, integration, and differential equations. Students should notice that these derivatives look very different from the derivatives of the original trig functions, which makes inverse trig a distinct topic rather than just an extension of basic trig rules.

Remember that
sin⁡−1x,cos⁡−1x,tan⁡−1x\sin^{-1}x, \cos^{-1}x, \tan^{-1}x
means arcsin(x), arccos(x), arctan(x) respectively. Both of these notations may be used, although arc.... is clearer.

In this unit

6 lessons

  1. 3.1The Chain RulePreview lesson
  2. 3.2Implicit DifferentiationPreview lesson
  3. 3.3Differentiating Inverse FunctionsPreview lesson
  4. 3.4Differentiating Inverse Trigonometric FunctionsThis lesson
  5. 3.5Selecting Procedures for Calculating DerivativesPreview lesson
  6. 3.6Calculating Higher-Order DerivativesPreview lesson

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