1. Derivatives of Natural Log Functions
Section 5.1
Calculus AP/Dual, Revised ©2017
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§5.1: Derivatives of Log Functions

2. Natural Logarithm Graph
Domain: 𝟎, ∞
Range: −∞, ∞
Continuous Function
One-to-One Function
Concaves Down
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§5.1: Derivatives of Log Functions

3. §5.1: Derivatives of Log Functions
Log Properties
𝐥𝐧 𝟏 =𝟎
𝐥𝐧 𝒂𝒃 = 𝐥𝐧 𝒂 +𝐥𝐧⁡𝒃
𝐥𝐧 𝒂 𝒏 =𝒏 𝐥𝐧⁡𝒂
𝐥𝐧 𝒂 𝒃 = 𝐥𝐧⁡𝒂 – 𝐥𝐧⁡𝒃
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§5.1: Derivatives of Log Functions

4. §5.1: Derivatives of Log Functions
Review Example 1
Expand 𝟑 𝐥𝐧 𝒙 𝟑 −𝟐 𝒙 𝟒
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§5.1: Derivatives of Log Functions

5. §5.1: Derivatives of Log Functions
Review Example 2
Condense 𝟐 [𝐥𝐧⁡𝒙 – ( 𝐥𝐧 𝒙+𝟏 +𝐥𝐧⁡(𝒙−𝟏))]
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§5.1: Derivatives of Log Functions

6. §5.1: Derivatives of Log Functions
Guess The Rule
Equation
Derivative
1) 𝒅 𝒅𝒙 𝐥𝐧 𝟒𝒙
2) 𝒅 𝒅𝒙 𝐥𝐧 𝟐 𝒙 𝟐
3) 𝒅 𝒅𝒙 𝐥𝐧 𝒙
4) 𝒅 𝒅𝒙 𝐥𝐧 𝟏+𝒙 𝟏+ 𝒙 𝟐 𝟐 𝟏+ 𝒙 𝟑 𝟑
5) 𝒅 𝒅𝒙 𝐥𝐧 (𝒖)
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§5.1: Derivatives of Log Functions

7. §5.1: Derivatives of Log Functions
Log Derivative Rules
𝒅 𝒅𝒙 𝐥𝐧⁡ 𝒖 = 𝒖 ′ 𝒖 ,𝒖>𝟎
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§5.1: Derivatives of Log Functions

8. Derivative of Natural Log
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§5.1: Derivatives of Log Functions

9. §5.1: Derivatives of Log Functions
Why?
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§5.1: Derivatives of Log Functions

10. §5.1: Derivatives of Log Functions
Steps
If necessary, use the log properties FIRST by EXPANDING
Apply the derivative
Simplify
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§5.1: Derivatives of Log Functions

11. §5.1: Derivatives of Log Functions
Example 1
Solve 𝒅𝒚 𝒅𝒙 if 𝒚= 𝐥𝐧⁡ 𝟐𝒙
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§5.1: Derivatives of Log Functions

12. §5.1: Derivatives of Log Functions
Example 1 (Easier way)
Solve 𝒅𝒚 𝒅𝒙 if 𝒚= 𝐥𝐧⁡ 𝟐𝒙
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§5.1: Derivatives of Log Functions

13. §5.1: Derivatives of Log Functions
Example 2
Solve for the derivative for 𝒇 𝒙 = 𝐥𝐧⁡ 𝒙 𝟐 +𝟏
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§5.1: Derivatives of Log Functions

14. §5.1: Derivatives of Log Functions
Example 3
Solve for the derivative 𝒇 𝒙 = (𝐥𝐧 𝒙) 𝟓 Hint: cannot use 𝒖′/𝒖 on this problem because of the exponent
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§5.1: Derivatives of Log Functions

15. §5.1: Derivatives of Log Functions
Example 4
Solve for the derivative : 𝒇 𝒙 = 𝐥𝐧 (𝒙 𝟒 )
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§5.1: Derivatives of Log Functions

16. §5.1: Derivatives of Log Functions
Your Turn
Solve for the derivative : 𝒇 𝒙 =𝒙 𝐥𝐧 𝒙
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§5.1: Derivatives of Log Functions

17. §5.1: Derivatives of Log Functions
Example 5
Solve for the derivative: 𝒇 𝒙 = 𝐥𝐧⁡ 𝒙 𝟐 −𝟑
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§5.1: Derivatives of Log Functions

18. §5.1: Derivatives of Log Functions
Example 6
Solve 𝒅𝒚 𝒅𝒙 if 𝒚= 𝒙 ∙𝐥𝐧⁡ 𝒙
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§5.1: Derivatives of Log Functions

19. §5.1: Derivatives of Log Functions
Example 7
Solve 𝒅𝒚 𝒅𝒙 if 𝒚=𝐥𝐧⁡ 𝒙+𝟏 𝒙−𝟏
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§5.1: Derivatives of Log Functions

20. §5.1: Derivatives of Log Functions
Example 7
Solve 𝒅𝒚 𝒅𝒙 if 𝒚=𝐥𝐧⁡ 𝒙+𝟏 𝒙−𝟏
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§5.1: Derivatives of Log Functions

21. §5.1: Derivatives of Log Functions
Your Turn
Solve 𝒅𝒚 𝒅𝒙 if 𝒚= 𝐥𝐧 𝒙 𝒙 𝟐 +𝟐
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§5.1: Derivatives of Log Functions

22. §5.1: Derivatives of Log Functions
Example 8
Solve 𝒅𝒚 𝒅𝒙 if 𝒚= 𝐥𝐧 𝐜𝐨𝐬 𝒙
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§5.1: Derivatives of Log Functions

23. §5.1: Derivatives of Log Functions
Your Turn
Solve 𝒅𝒚 𝒅𝒙 if 𝒚= 𝐥𝐧 𝐜𝐬𝐜 𝒙
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§5.1: Derivatives of Log Functions

24. AP Multiple Choice Practice Question 1 (non-calculator)
Solve 𝒅𝒚 𝒅𝒙 for 𝒚= 𝐥𝐧 𝒙 𝟐 +𝟒 (A) 𝟏 𝒙 (B) 𝒙 𝒙 𝟐 +𝟒 (C) 𝒙 𝒙 𝟐 +𝟒 (D) 𝟐𝒙 𝒙 𝟐 +𝟒
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§5.1: Derivatives of Log Functions

25. AP Multiple Choice Practice Question 1 (non-calculator)
Solve 𝒅𝒚 𝒅𝒙 for 𝒚= 𝐥𝐧 𝒙 𝟐 +𝟒
Vocabulary
Connections and Process
Answer and Justifications
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§5.1: Derivatives of Log Functions

26. §5.1: Derivatives of Log Functions
Assignment
Page 325
5-8 all, odd (omit 55), 64, 65A, 73, 75
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§5.1: Derivatives of Log Functions