Lesson 3 8 Derivative of Natural Logs And

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: loga x = y ay = x Cancellation Equations: loga (ax) = x a loga x = x x is a real number 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)

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

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

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

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

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 d 1 --- (ln |x|) = ------ (from example 6 in the book) dx x

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

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

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