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: 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)

5. 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)

6. 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

7. 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

8. 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

9. 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

10. 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)

11. 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)

12. 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

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