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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