The Natural Logarithmic

FunctionDifferentiation

Integration

Properties of the Natural Log Function

• If a and b are positive numbers and n is rational, then the following properties are true:

ln(1) 0

ln( ) ln lnab a b

ln( ) lnna n a

ln ln lna

a bb

The Algebra of Logarithmic Expressions

10ln ln10 ln 9

9

1/ 2 1ln 3 2 ln 3 2 ln 3 2

2x x x

6ln ln 6 ln 5

5

xx

22

3 2

3ln

1

x

x x

The Derivative of the Natural Logarithmic Function

Let u be a differentiable function of x

1ln , 0

dx x

dx x

ln , 0d u

u udu u

Differentiation of Logarithmic Functions

ln 2d

xdx

2ln 1d

xdx

lndx x

dx 3ln

dx

dx

Differentiation of Logarithmic Functions

ln 2d

xdx

2 2u x u

ln , 0d u

u udu u

2 1

2

u

u x x

Differentiation of Logarithmic Functions

2ln 1d

xdx

2 1 2u x u x

ln , 0d u

u udu u

2

2

1

u x

u x

Differentiation of Logarithmic Functions

ln APPLY PRODUCT RULE!!!!!!!!!!dx x

dx

11 ln x x

x

ln 1x

ln ln lnd d dx x x x x x

dx dx dx

Differentiation of Logarithmic Functions

3ln CHAIN RULE!!!

dx

dx

23 ln ln

dx xdx

23 ln x

x

2 13 ln x

x

Logarithmic Properties as Aids to Differentiation

• Differentiate:

ln 1f x x

1/2 1ln 1 ln 1

2f x x x

11 1

2 1 2 1

xf x

x x

Logarithmic Properties as Aids to Differentiation

• Differentiate:

22

3

1ln

2 1

x xf x

x

Logarithmic Differentiation

• Differentiate:

This can get messy with the quotient or product and chain rules. So we will use ln rules to help simplify this and apply implicit differentiation and then we solve for y’…

2

2

2

1

xf x

x

Derivative Involving Absolute Value

• Recall that the ln function is undefined for negative numbers, so we often see expressions of the form ln|u|. So the following theorem states that we can differentiate functions of the form y= ln|u| as if the absolute value symbol is not even there.

• If u is a differentiable function such that u≠0 then:

lnd u

udu u

Derivative Involving Absolute Value

• Differentiate: ln cosf x x

ln cosd u

xdx u

cos , sinu

u x u xu

sin

cos

u x

u x

Finding Relative Extrema

• Locate the relative extrema of • Differentiate:

• Set = 0 to find critical points=02x+2=0X=-1, Plug back into original to find yy=ln(1-2+3)=ln2 So, relative extrema is at (-1, ln2)

Homework• 5.1 Natural Logarithmic Functions and the

Number e Derivative #19-35,47-65, 71,79,93-96

General Power Rule for Integration

• Recall that it has an important disclaimer- it doesn’t apply when n = -1. So we can not integrate functions such as f(x)=1/x.

• So we use the Second FTC to DEFINE such a function.

Integration Formulas• Let u be a differentiable function of x

1lndx x c

x

1lndx u c

u

Using the Log Rule for Integration

2dxx

1Factor out the constant:2

This gives us a form we recognize and can easily integrate.

dxx

2ln x c

2Use rules to clean things up:ln x c

Using the Log Rule with a Change of Variables

( 14 𝑥−1 )𝑑𝑥 Let u=4x-1, so du=4dx

and dx=

1ln

4u c

1ln 4 1

4x c

Finding Area with the Log Rule

• Find the area of the region bounded by the graph of y, the x-axis and the line x=3.

3 3

20 0Set up your integral: ( )

1

xy x dx dx

x

We have an integral in the form

u

u

2 1 so 2Let u x u xdx

2

1 1 1 12

2 1 2xdx du

x u

3

2

0

1 1ln ln 1

2 2u x

2 21 1 1 1 1ln 3 1 ln 0 1 ln10 ln 1 ln10 ln 10

2 2 2 2 2

2 1

xy x

x

Recognizing Quotient Forms of the Log Rule

23 3

3

3 1ln

xdx u x x x x c

x x

2sectan ln tan

tan

xdx u x x c

x

1/31 1 3 13 2 ln 3 2 ln 3 2

3 2 3 3 2 3dx u x dx x c x c

x x

DefinitionThe natural logarithmic function is defined

by

The domain of the natural logarithmic function is the set of all positive real numbers

1

1ln , 0

xx dt x

t

u-Substitution and the Log Rule1

_ _ln

dya differential equation

dx x x

1_ _ :

lnIntegrate both sides y dx

x x

1lnu x du dx

x

uduu

ln u c

ln ln x c

Long Division With Integrals

How you know it’s long Division

• If it is top heavy that means it is long division.o Example

3 2

2

4 4 96 100

25

x x x

x

Example 12 5 6

5

x x

x

25 5 6x x x

2

2

6

5

5 5 6

-x 5

xx

x x x

x

Continue Example 1

216ln 5

2x x

6

5xx

Example 22 3 2

1

x x

x

21 3 2x x x

Continue Example 22

2

21 3 2

-x 1

2 2

-2x -2

xx x x

x

x

2x 212

2x x

Using Long Division Before Integrating

2

2

1

1

x xdx

x

22 2

2 2

11

1 1 11 1

rxx x x

dx x x xx x

21

1

xdx

x

2 1

xdx dx

x

21ln 1

2x x c

2: 1 0, ... _ _ _ _ _Note x x so no need for absolute value

Using a Trigonometric Identity

tan xdx

Guidelines for integration

1. Learn a basic list of integration formulas. (including those given in this section, you now have 12 formulas: the Power Rule, the Log Rule, and ten trigonometric rules. By the end of section 5.7 , this list will have expanded to 20 basic rules)

2. Find an integration formula that resembles all or part of the integrand, and, by trial and error, find a choice of u that will make the integrand conform to the formula.

3. If you cannot find a u-substitution that works, try altering the integrand. You might try a trigonometric identity, multiplication and division by the same quantity, or addition and subtraction of the same quantity. Be creative.

4. If you have access to computer software that will find antiderivatives symbolically, use it.

Integrals of the Six Basic Trigonometric Functions

sin cosudu u c cos sinudu u c tan ln cosudu u c cot ln sinudu u c

sec ln sec tanudu u u c csc ln csc cotudu u u c

Homework• 5.2 Log Rule for Integration and Integrals for Trig

Functions (substitution) #1-39, 47-53,  67