Derivative of Natural Logs And Logarithmic Differentiation Lesson 3-9 Derivative of Natural Logs And Logarithmic Differentiation

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

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)

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)

Laws of Logs Practice Simplify the following equations using laws of logarithms y = ln (12a4 / 5b3) y = (ln12 + 4lna) – (ln5 + 3lnb) = ln12 + 4lna – ln5 – 3lnb y = ln(2a4b7c3) y = ln2 + 4lna + 7lnb + 3lnc

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)]

Laws of Logs Practice Y = ln a – ln b + ln c Y = 7ln a + 3ln b Combine into a single expression using laws of logarithms Y = ln a – ln b + ln c Y = ln (ac/b) Y = 7ln a + 3ln b Y = ln a^2b^3 Y = 3ln a – 5ln c Y = ln (a^3/b^5)

Derivatives of Logarithmic Functions --- (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

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)

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

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)

Summary & Homework Summary: Derivative of Derivatives Use all known rules to find higher order derivatives