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2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

Part of Differentiation: Definition and Basic Derivative Rules in AP® Calculus BC. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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The remaining basic trig derivatives can be found by rewriting the functions in terms of sine and cosine and then using earlier derivative rules. The key results are
ddx(tan⁡x)=sec⁡2x\frac{d}{dx}(\tan x)=\sec^2 x
,
ddx(cot⁡x)=−csc⁡2x\frac{d}{dx}(\cot x)=-\csc^2 x
,
ddx(sec⁡x)=sec⁡xtan⁡x\frac{d}{dx}(\sec x)=\sec x\tan x
,
and
ddx(csc⁡x)=−csc⁡xcot⁡x\frac{d}{dx}(\csc x)=-\csc x\cot x
These are important because they expand the list of trig functions students can differentiate directly, and they appear often in later units involving identities, implicit differentiation, and applications. Each of these derivatives must be used with attention to domain, since the original trig functions are not defined everywhere.

In this unit

10 lessons

  1. 2.1Defining Average and Instantaneous Rates of Change at a PointPreview lesson
  2. 2.2Defining the Derivative of a Function and Using Derivative NotationPreview lesson
  3. 2.3Estimating Derivatives of a Function at a PointPreview lesson
  4. 2.4Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not ExistPreview lesson
  5. 2.5Applying the Power RulePreview lesson
  6. 2.6Derivative Rules: Constant, Sum, Difference, and Constant MultiplePreview lesson
  7. 2.7Derivatives of cos x, sin x, eˣ, and ln xPreview lesson
  8. 2.8The Product RulePreview lesson
  9. 2.9The Quotient RulePreview lesson
  10. 2.10Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant FunctionsThis lesson

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