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Lesson 3-8

Lesson 3-8. Derivative of Natural Logs And Logarithmic Differentiation. Objectives. Know derivatives of regular and natural logarithmic functions Take derivatives using logarithmic differentiation. Vocabulary. None new. Logarithmic Functions. Logarithmic Functions:

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Lesson 3-8

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  1. Lesson 3-8 Derivative of Natural Logs And Logarithmic Differentiation

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

  3. Vocabulary • None new

  4. Logarithmic Functions Logarithmic Functions: loga x = y ay = x Cancellation Equations: loga (ax) = x x is a real number a loga x = x x > 0 Laws of Logarithms: loga (xy) = loga x + loga y loga (x/y) = loga x - loga y loga xr = r loga x (where r is a real number)

  5. Natural Logs Natural Logarithms: loge x = ln x ln e = 1 ln x = y ey = x Cancellation Equations: ln (ex) = x ln e = x x is a real number eln x = x x > 0 Change of Base Formula: loga x = (ln x) / (ln a)

  6. Laws of Logs Practice Simplify the following equations using laws of logarithms • y = ln (12a4 / 5b3) • y = ln(2a4b7c3)

  7. Laws of Logs Practice Simplify the following equations using laws of logarithms • y = ln[(x²)5(3x³)4 / ((x + 1)³(x - 1)²)] • f(x) = ln[(tan3 2x)(cos4 2x) / (e5x)]

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

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

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

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

  12. Example 3 Find the derivatives of the following: 5. f(x) = x²ln(x) 6. f(x) = log2(x² + 1) Product Rule! f’(x) = x²(1/x) + 2x ln (x) = x + 2x ln (x) Log base a Rule! d u’ --- (loga u) = ----------- dx u ln a f’(x) = (2x) / (x² + 1)(ln 2)

  13. 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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