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Lesson 3-8 Derivative of Natural Logs And Logarithmic Differentiation

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Objectives Know derivatives of regular and natural logarithmic functions Take derivatives using logarithmic differentiation

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Vocabulary None new

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Logarithmic Functions Logarithmic Functions: log a x = y a y = x Cancellation Equations: log a (a x ) = x x is a real number a log a x = x x > 0 Laws of Logarithms: log a (xy) = log a x + log a y log a (x/y) = log a x - log a y log a x r = r log a x (where r is a real number)

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Natural Logs Natural Logarithms: log e x = ln x ln e = 1 ln x = y e y = x Cancellation Equations: ln (e x ) = x ln e = x x is a real number e ln x = x x > 0 Change of Base Formula: log a x = (ln x) / (ln a)

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Laws of Logs Practice 1.y = ln (12a 4 / 5b 3 ) 2.y = ln(2a 4 b 7 c 3 ) Simplify the following equations using laws of logarithms

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Laws of Logs Practice 3.y = ln[(x²) 5 (3x³) 4 / ((x + 1)³(x - 1)²)] 4.f(x) = ln[(tan 3 2x)(cos 4 2x) / (e 5x )] Simplify the following equations using laws of logarithms

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Laws of Logs Practice Y = ln a – ln b + ln c Y = 7ln a + 3ln b Y = 3ln a – 5ln c Combine into a single expression using laws of logarithms

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Derivatives of Logarithmic Functions d 1 --- (log a x) = -------- dx x ln a d d 1 1 --- (log e x) = ---(ln x) = -------- = ---- dx dx x ln e x d 1 du u' --- (ln u) = -------- = ------- Chain Rule dx u d 1 --- (ln |x|) = ------ (from example 6 in the book) dx x

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Example 1 1. f(x) = ln(2x) 2. f(x) = ln(√x) Find second derivatives of the following: f’(x) = 2/2x f’(x) = 1/x f’(x) = ½ (x -½ ) / x = 1 / (2 x x) = 1/2x u = 2x du/dx = 2 d(ln u)/dx = u’ / u u = x du/dx = ½ x -½ d(ln u)/dx = u’ / u f(x) = ½ (ln x) f’(x) = 1/(2x)

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Example 2 3. f(x) = ln(x² – x – 2) 4. f(x) = ln(cos x) f’(x) = (2x – 1) / (x² – x – 2) f’(x) = (-sin x) / (cos x) f’(x) = - tan x u = (x² – x – 2) u’ = (2x – 1) u = (cos x) u’ = (-sin x)

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Example 3 5. f(x) = x²ln(x) 6. f(x) = log 2 (x² + 1) Find the derivatives of the following: f’(x) = (2x) / (x² + 1)(ln 2) f’(x) = x²(1/x) + 2x ln (x) = x + 2x ln (x) Product Rule! Log base a Rule! d u’ --- (log a u) = ----------- dx u ln a

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Summary & Homework Summary: –Derivative of Derivatives –Use all known rules to find higher order derivatives Homework: –pg 240 - 242: 5, 9, 17, 18, 25, 29, 49, 57

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