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

bc

bccb log

loglnlnlog

Review Laws of Logs

• Algebraic Properties of Logarithms

• Remember that means .xy blog xb y

Review Laws of Logs

• Algebraic Properties of Logarithms

• Remember that means .• Logarithmic and exponential functions are inverse

functions.

xy blog xb y

xby xy blog

xb bx logxbbx log

ybx yx blog

Derivatives of Logs

• We will start this definition with another way to express e. In chapter 2, we defined e as:

• Now, we will look at e as:

• We make the substitution v = 1/x, and we know that as

ex

x

x

11lim

ev v

v

/1

0)1(lim

x 0v

Defintion

hxhxx

dxd

h

ln)ln(lim][ln0

Defintion

hxhxx

dxd

h

ln)ln(lim][ln0

x

hxhhln1lim

0

Defintion

hxhxx

dxd

h

ln)ln(lim][ln0

x

hxhhln1lim

0

xh

hh1ln1lim

0

Definition

• We will now let v=h/x, so h = vx

xh

hh1ln1lim

0

)1ln(1lim0

vvxv

)1ln(1lim10

vvx v

Definition

• Finally

v

vv

x/1

0)1ln(lim1

])1(limln[1 /1

0

v

vv

x

)1ln(1lim10

vvx v

exln1

xx

dxd 1][ln

Defintion

• Now we will look at the derivative of a log with any base.

][log xdxd

b

Defintion

• Now we will look at the derivative of a log with any base.

• We will use the change of base formula to rewrite this as

][log xdxd

b

bx

dxdlnln

Defintion

• Now we will look at the derivative of a log with any base.

• We will use the change of base formula to rewrite this as

][log xdxd

b

bxx

dxd

bbx

dxd

ln1][ln

ln1

lnln

Definition

• In summary:

xx

dxd 1][ln

bxx

dxd

b ln1][log

dxdu

uu

dxd

1][ln

dxdu

buu

dxd

b ln1][log

Example 1

• The figure below shows the graph of y = lnx and its tangent lines at x = ½, 1, 3, and 5. Find the slopes of the tangent lines.

Example 1

• The figure below shows the graph of y = lnx and its tangent lines at x = ½, 1, 3, and 5. Find the slopes of the tangent lines.

• Since the derivative of y = lnx is dy/dx = 1/x, the slopes of the tangent lines are: 2, 1, 1/3, 1/5.

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 (the derivative) will never equal zero, so there are no horizontal tangent lines.

• As the value of x approaches infinity, the slope of the tangent line does approach 0, but never gets there.

Example 2

• Find )]1[ln( 2 xdxd

Example 2

• Find

• We will use a u-substitution and let

)]1[ln( 2 xdxd

12 xu

xdxdu 2

uu

dxd 1][ln

12)]1[ln( 2

2

xxx

dxd

Example 3

• Find

xxx

dxd

1sinln

2

Example 3

• Find

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

xxx

dxd

1sinln

2

)1ln(21)ln(sinln2

1sinln

2

xxxdxd

xxx

dxd

Example 3

• Now, we solve.

)1ln(

21)ln(sinln2 xxx

dxd

)1(21

sincos2

xxx

x

xx

x 221cot2

Absolute Value

• Lets look at |]|[ln xdxd

Absolute Value

• Lets look at

• If x > 0, |x| = x, so we have

|]|[ln xdxd

xx

dxdx

dxd 1][ln|]|[ln

Absolute Value

• Lets look at

• If x > 0, |x| = x, so we have

• If x < 0, |x|= -x, so we have

|]|[ln xdxd

xx

dxdx

dxd 1][ln|]|[ln

xxx

dxdx

dxd 11][ln|]|[ln

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 xdxd

xx

dxdx

dxd 1][ln|]|[ln

xxx

dxdx

dxd 11][ln|]|[ln

xx

dxd 1|]|[ln

Logarithmic Differentiation

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

• Find the derivative of

42

32

)1(147

xxxy

Logarithmic Differentiation

• Find the derivative of

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

42

32

)1(147

xxxy

)1ln(4)147ln(31ln2ln 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

xxxy

)1ln(4)147ln(31ln2ln 2xxxy

218

)147(3721

xx

xxdxdy

y

Logarithmic Differentiation

• Find the derivative of 42

32

)1(147

xxxy

218

)147(3721

xx

xxdxdy

y

42

32

2 )1(147

18

6312

xxx

xx

xxdxdy

Homework

• Section 3.2• 1-29 odd• 35, 37