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Fundamental Properties of Differentiation

Michael Green

Michael Green

6 min read

Next Topic - The Chain Rule

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Study Guide Overview

This study guide covers derivatives of tangent and reciprocal trigonometric functions. It includes the derivative of tan(x) and tan(kx) using the quotient and chain rules. It also defines and derives the derivatives of sec(x), csc(x), and cot(x). The guide provides worked examples, practice questions, a glossary, and exam strategies.

#Study Notes: Derivatives of Trigonometric Functions

#Table of Contents

  1. Derivative of the Tangent Function
    • What is the Derivative of tan⁡x\tan xtanx?
    • What is the Derivative of tan⁡kx\tan kxtankx?
    • Worked Examples
  2. Derivatives of Reciprocal Trigonometric Functions
    • What are the Reciprocal Trig Functions?
    • What are the Derivatives of the Reciprocal Trig Functions?
    • How to Derive the Derivatives
    • Worked Example
  3. Practice Questions
  4. Glossary
  5. Summary and Key Takeaways
  6. Exam Strategy

#Derivative of the Tangent Function

#What is the Derivative of tan⁡x\tan xtanx?

Key Concept

If f(x)=tan⁡xf(x) = \tan xf(x)=tanx, then the derivative is given by: f′(x)=sec⁡2xf'(x) = \sec^2 xf′(x)=sec2x

This can be shown using the identity tan⁡x≡sin⁡xcos⁡x\tan x \equiv \frac{\sin x}{\cos x}tanx≡cosxsinx​ and the quotient rule.

The quotient rule states that if y=uvy = \frac{u}{v}y=vu​, then y′=u′v−uv′v2y' = \frac{u'v - uv'}{v^2}y′=v2u′v−uv′​

Let u=sin⁡xu = \sin xu=sinx and v=cos⁡xv = \cos xv=cosx. - u′=cos⁡xu' = \cos xu′=cosx - v′=−sin⁡xv' = -\sin xv′=−sinx

Applying the quotient rule: y′=cos⁡x⋅cos⁡x+sin⁡x⋅sin⁡xcos⁡2xy' = \frac{\cos x \cdot \cos x + \sin x \cdot \sin x}{\cos^2 x}y′=cos2xcosx⋅cosx+sinx⋅sinx​ Simplifying: y′=cos⁡2x+sin⁡2xcos⁡2xy' = \frac{\cos^2 x + \sin^2 x}{\cos^2 x}y′=cos2xcos2x+sin2x​ Using the identity sin⁡2x+cos⁡2x≡1\sin^2 x + \cos^2 x \equiv 1sin2x+cos2x≡1: y′=1cos⁡2x=sec⁡2xy' = \frac{1}{\cos^2 x} = \sec^2 xy′=cos2x1​=sec2x

#What is the Derivative of $\tan k...

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Previous Topic - The Quotient RuleNext Topic - The Chain Rule

Question 1 of 11

What is the derivative of f(x)=tan⁡4xf(x) = \tan 4xf(x)=tan4x ? 🚀

4 \sec^2 4x

sec⁡24x\sec^2 4xsec24x

4 \tan 4x \sec 4x

4 \cot 4x