The Natural Logarithmic Function: Differentiation

(5.1)

The Natural Logarithmic Function: Differentiation

(5.1)February 21st, 2013February 21st, 2013

I. The Natural Logarithmic FunctionI. The Natural Logarithmic Function

Def. of the Natural Logarithmic Function: The natural logarithmic function is defined by .

The domain is the set of all positive real numbers.

ln x =1tdt,x> 0

1

x

∫

Thm. 5.1: Properties of the Natural Logarithmic Function:1. Domain: , Range:2. The function is continuous, increasing, and one-to-one3. The graph is concave downward

0,∞( ) −∞,∞( )

Thm. 5.2: Logarithmic Properties:1.

2.

3.

4.

ln 1( ) =0

ln ab( ) =lna+ lnb

ln an( ) =nlna

lna

b⎛⎝⎜

⎞⎠⎟=lna−lnb

Ex. 1: Use properties of logarithms to expand the following logarithmic expressions.

a.

b.

ln8x

3

ln4x +1( )

3

2x−1

II. The Number eII. The Number e

*Recall that the base of the natural logarithm is the number , so .

Def. of e: The letter e denotes the positive real number such that .

e≈2.718

lne=1tdt=1

1

e

∫

ln x =loge x

III. The Derivative of the Natural LogarithmIII. The Derivative of the Natural Logarithm

Thm. 5.3: Derivative of the Natural Logarithmic Function: Let u be a differentiable function of x.

1. (since )

2.

d

dxln x[ ] =

1x,x> 0

d

dxln x[ ] =

ddx

1tdt=

1x1

x

∫

d

dxlnu[ ] =

1u

dudx

=u'u

,u> 0

Ex. 2: Differentiate each function.

a.

b.

c.

d.

e.

f.

f (x)=ln(5x)

f (x)=ln(x3 −4)

f (x)=x2 lnx

f (x)=12(lnx)4

f (x)=ln x2 −1

f (x)=lnx2 (4x−1)3

x+6

*We can use logarithmic differentiation to differentiate nonlogarithmic functions.

Ex. 3: Use logarithmic differentiation to find the derivative of .y=

(x+1)2

4x2 +1,x≠−1

You Try: Use logarithmic differentiation to find the derivative of .y= (x−2)(x2 −1)

Thm. 5.4: Derivative Involving Absolute Value: If u is a differentiable function of x such that , then .

u ≠0d

dxln u⎡⎣ ⎤⎦=

u'u

Ex. 4: Find the derivative of .f (x)=lnsinx

Ex. 5: Find the relative extrema of .

y=ln(x2 + 4)