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