STANDARD: STUDENTS SHOULD BE ABLE TO USE DERIVATIVES TO SOLVE A VARIETY OF PROBLEMS

OBJECTIVE: DERIVE AND APPLY THE DERIVATIVE OF A LOGARITHM

Agenda:

1. Review of Logarithms

2. Determine the derivative of a Logarithm

3. Notes on 3.5

4. Practice Problems on Logarithm Derivatives

5. Homework questions on Inverse Derivatives

DERIVAT

IVES O

F

LOGARIT

HMIC F

UNCTIONS

SE

CT

I ON

3. 6

THE PROOF

log

' ln '

''

ln'

'ln

'log

ln

ya

y

y

a

y u a u

a y a u

uy

a au

yu a

d uso u

dx u a

ANOTHER PROOF!!!!!

log

' '

''

''

'ln

ye

y

y

y u e u

e y u

uy

eu

yu

d uso u

dx u

EXAMPLES:

1. Find the derivative of 2

4log xy e

2

2

' 2

x

x

u e

u e x

2

2

2 2'

ln 4ln 4

x

x

xe xy

e

2. Find the derivative of 2ln( 4)y x

2 4

' 2

u x

u x

2

2'

4

xy

x

“PROPERTIES OF LOGARITHMS”

. log( ) log log

. log log log

. log logb

i a b a b

aii a b

b

iii a b a

3. Given find

Using your log properties:

2 3

10log 3 4 2 1y x x 'y

2 3

10 10

10 10

log 3 4 log 2 1

2log 3 4 3log 2 1

y x x

y x x

3 4

' 3

u x

u

2 1

' 2

u x

u

3 22 33 4 ln10 2 1 ln10

6 6'

3 4 ln10 2 1 ln10

x x

so yx x

HOW HARD IS THIS PROBLEM WITHOUT USING THE LOG PROPERTIES?

Find the derivative of

2

3

2

3

3 2 4 6

5 4

1ln 2ln 3 2 ln 4 6 3ln 5 4

2' 3 1 4 52 33 2 2 4 6 5 4

6 2 15'

3 2 4 6 5 4

3 2 4 66 2 15'

3 2 4 6 5 4 5 4

x xy

x

y x x x

y

y x x x

y yx x x

x xy

x x x x

ONE LAST EXAMPLE!!!!

Find the derivative of

xy x

ln ln

' 11 ln

' ln 1 x

y x x

yx x

y x

y x x