Derivatives of Natural Log Functions Section 5.1 Calculus AP/Dual, Revised ©2017 viet.dang@humbleisd.net 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

Natural Logarithm Graph Domain: 𝟎, ∞ Range: −∞, ∞ Continuous Function One-to-One Function Concaves Down 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

§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 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

§5.1: Derivatives of Log Functions Review Example 1 Expand 𝟑 𝐥𝐧 𝒙 𝟑 −𝟐 𝒙 𝟒 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

§5.1: Derivatives of Log Functions Review Example 2 Condense 𝟐 [𝐥𝐧⁡𝒙 – ( 𝐥𝐧 𝒙+𝟏 +𝐥𝐧⁡(𝒙−𝟏))] 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

§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) 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

§5.1: Derivatives of Log Functions Log Derivative Rules 𝒅 𝒅𝒙 𝐥𝐧⁡ 𝒖 = 𝒖 ′ 𝒖 ,𝒖>𝟎 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

Derivative of Natural Log http://www.math.uri.edu/~bkaskosz/flashmo/derplot/ 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

§5.1: Derivatives of Log Functions Why? 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

§5.1: Derivatives of Log Functions Steps If necessary, use the log properties FIRST by EXPANDING Apply the derivative Simplify 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

§5.1: Derivatives of Log Functions Example 1 Solve 𝒅𝒚 𝒅𝒙 if 𝒚= 𝐥𝐧⁡ 𝟐𝒙 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

§5.1: Derivatives of Log Functions Example 1 (Easier way) Solve 𝒅𝒚 𝒅𝒙 if 𝒚= 𝐥𝐧⁡ 𝟐𝒙 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

§5.1: Derivatives of Log Functions Example 2 Solve for the derivative for 𝒇 𝒙 = 𝐥𝐧⁡ 𝒙 𝟐 +𝟏 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

§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 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

§5.1: Derivatives of Log Functions Example 4 Solve for the derivative : 𝒇 𝒙 = 𝐥𝐧 (𝒙 𝟒 ) 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

§5.1: Derivatives of Log Functions Your Turn Solve for the derivative : 𝒇 𝒙 =𝒙 𝐥𝐧 𝒙 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

§5.1: Derivatives of Log Functions Example 5 Solve for the derivative: 𝒇 𝒙 = 𝐥𝐧⁡ 𝒙 𝟐 −𝟑 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

§5.1: Derivatives of Log Functions Example 6 Solve 𝒅𝒚 𝒅𝒙 if 𝒚= 𝒙 ∙𝐥𝐧⁡ 𝒙 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

§5.1: Derivatives of Log Functions Example 7 Solve 𝒅𝒚 𝒅𝒙 if 𝒚=𝐥𝐧⁡ 𝒙+𝟏 𝒙−𝟏 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

§5.1: Derivatives of Log Functions Example 7 Solve 𝒅𝒚 𝒅𝒙 if 𝒚=𝐥𝐧⁡ 𝒙+𝟏 𝒙−𝟏 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

§5.1: Derivatives of Log Functions Your Turn Solve 𝒅𝒚 𝒅𝒙 if 𝒚= 𝐥𝐧 𝒙 𝒙 𝟐 +𝟐 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

§5.1: Derivatives of Log Functions Example 8 Solve 𝒅𝒚 𝒅𝒙 if 𝒚= 𝐥𝐧 𝐜𝐨𝐬 𝒙 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

§5.1: Derivatives of Log Functions Your Turn Solve 𝒅𝒚 𝒅𝒙 if 𝒚= 𝐥𝐧 𝐜𝐬𝐜 𝒙 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

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 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

AP Multiple Choice Practice Question 1 (non-calculator) Solve 𝒅𝒚 𝒅𝒙 for 𝒚= 𝐥𝐧 𝒙 𝟐 +𝟒 Vocabulary Connections and Process Answer and Justifications 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions

§5.1: Derivatives of Log Functions Assignment Page 325 5-8 all, 41-63 odd (omit 55), 64, 65A, 73, 75 12/27/2018 6:04 PM §5.1: Derivatives of Log Functions