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:
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
1. y = ln (12a4 / 5b3)
2. y = ln(2a4b7c3)
Simplify the following equations using laws of logarithms
Laws of Logs Practice
3. y = ln[(x²)5(3x³)4 / ((x + 1)³(x - 1)²)]
4. f(x) = ln[(tan3 2x)(cos4 2x) / (e5x)]
Simplify the following equations using laws of logarithms
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
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 Ruledx u dx u
d 1--- (ln |x|) = ------ (from example 6 in the book)dx x
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 xx)
= 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)
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)
Example 3
5. f(x) = x²ln(x)
6. f(x) = log2(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’--- (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
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