Derivatives of Logarithmic Functions

Objective: Obtain derivative formulas for logs.

Review Laws of Logs

• Algebraic Properties of Logarithms

1. Product Property

2. Quotient Property

3. Power Property

4. Change of base

caac bbb loglog)(log

caca bbb loglog)/(log

ara br

b log)(log

b

c

b

ccb log

log

ln

lnlog

Definitions to Remember

xx

dx

d 1][ln

bxx

dx

db ln

1][log

dx

du

uu

dx

d

1ln

dx

du

buu

dx

db

ln

1log

Example 1

• Does the graph of y = lnx have any horizontal tangents?

Example 1

• Does the graph of y = lnx have any horizontal tangents?

• The answer is no. 1/x will never equal zero, so there are no horizontal tangent lines.

Example 2

• Find )]1[ln( 2 xdx

d

1

2)]1[ln(

22

x

xx

dx

d

Example 3• Find xxy

dx

dyln if 2

Absolute Value

• Lets look at

• If x > 0, |x| = x, so we have

• If x < 0, |x|= -x, so we have

• So we can say that

|]|[ln xdx

d

xx

dx

dx

dx

d 1][ln|]|[ln

xxx

dx

dx

dx

d 11][ln|]|[ln

xx

dx

d 1|]|[ln

Example 4• Find 1 if xy

dx

dy

Example 5

• Find

xxx

dx

d

1

sinln

2

Example 5

• Find

• We will use our rules of logs to make this a much easier problem.

xxx

dx

d

1

sinln

2

)1ln(2

1)ln(sinln2

1

sinln

2

xxxdx

d

x

xx

dx

d

Example 5

• Now, we solve.

)1ln(

2

1)ln(sinln2 xxx

dx

d

)1(2

1

sin

cos2

xx

x

x

xx

x 22

1cot

2

Logarithmic Differentiation

• This is another method that makes finding the derivative of complicated problems much easier.

• Find the derivative of

42

32

)1(

147

x

xxy

Logarithmic Differentiation

• Find the derivative of

• First, take the natural log of both sides and treat it like example 3.

42

32

)1(

147

x

xxy

)1ln(4)147ln(3

1ln2ln 2xxxy

Logarithmic Differentiation

• Find the derivative of

• First, take the natural log of both sides and treat it like example 3.

42

32

)1(

147

x

xxy

)1ln(4)147ln(3

1ln2ln 2xxxy

21

8

)147(3

721

x

x

xxdx

dy

y

Logarithmic Differentiation

• Find the derivative of 42

32

)1(

147

x

xxy

21

8

)147(3

721

x

x

xxdx

dy

y

42

32

2 )1(

147

1

8

63

12

x

xx

x

x

xxdx

dy

Homework

• Pages 247-248• 1-33 odd