5-1: natural logs and differentiation objectives: ©2003roy l. gover () review properties of natural...
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5-1: Natural Logs and Differentiation
Objectives:
©2003Roy L. Gover (www.mrgover.com)
•Review properties of natural logarithms•Differentiate natural logarithm functions
Review
ln yx y e x
n logl ex x
2.718281828e
ExampleWrite as a natural logarithm:
2 7.3891e 1
2 1 87 .64e
ExampleWrite in exponential form:
ln 7.3891 2
1ln1.6487
2
Properties of Logarithms
ln(1)=0
ln(ab)=ln a + ln b
ln an=n ln a
ln ln lna
a bb
ln e=1
•What is the domain?
•What is the range?
Important Idea
•Wassup at x=1?
The graph of the natural log function looks like:
Try ThisExpand the log function:
ln[( ( )] )x y x y
ln( ) ln( )x y x y
Try ThisExpand the log function:
10ln
3
x
y
ln(10 ) ln(3 )x y then:
ln10 ln ln 3 lnx y
Try ThisExpand the log function:
ln 2x
1ln(2 )
2x then:
1(ln 2 ln )
2x
Try ThisEvaluate:
4ln e
4ln 4(1) 4e
Try ThisEvaluate:
2 2ln ( )e x y
2 2x y
Try ThisEvaluate:
ln 5
1.6094
Try ThisWrite as a logarithm of a single quantity:
1ln 2ln
2x y
2ln x y
Try ThisWrite as a logarithm of a single quantity:
n(3 1) ln 2l (2 )x x
3 1ln
2 2
x
x
Try ThisFind the antiderivative:1
dxx
Can’t solve using the power rule
Important Idea
•There exists an area
under the curve
equal to 1
1( )f x
x
ln 1e •
• 1b
adx
x is an area under
1( )f x
x
Definition
1 2.72e
1( )f x
xe is the
positive real number such that:
1
1ln 1
ee dt
t
Area = 1
Definitionfrom the previous definition...
1 1[ln ]
x
a
d dx dt
dx dx t x
therefore:
1[ln ]
dx
dx x
memorize
The chain rule version:
1[ln ]
d duu
dx u dx
Definition
Examples
[ln 4 ] d
xdx
2[ln(3 1)]d
xdx
Try This
3[ln( 1 ] 4 )d
xdx
2
3
12
4 1
x
x
Example
[ ln ]d
x xdx
Hint: use the product rule
Example
lnd x
dx x
Hint: use the quotient rule
Example
ln1
d x
dx x
Is this a quotient rule problem?
Example
2[ln ]d
xdx
2[(ln ) ]d
xdx
Try This
2 3( ) 2(ln )f x x x
Find the derivative:
Hint: Rewrite using log properties then use chain rule
Solution
3
3 3
3
2(2 ln )
2(2 )(ln )
= 16(ln )
x x
x x
x x
Rewrite:2 3( ) 2(ln )f x x x
SolutionUse chain rule:
2
2
1'( ) 1 16(3)(ln )
48(ln ) =
1
f x xx
x
x
Try This
3
2 2
2 1( ) ln
( 1)
xf x
x x
Rewrite using log properties before differentiation...
3 2 2
3 2 2
3 2
( ) ln 2 1 ln[ ( 1) ]
1 ln(2 1) [ln ln( 1) ]
21
ln(2 1) ln 2ln( 1)2
f x x x x
x x x
x x x
Rewrite:Solution
23 2
1 1 1 1 '( ) (6
) 2 (2 )2 2 1 1
f x x xxx x
…then differentiate
Solution
2
3 2
3 1 4
2 1 1
x x
xx x
And simplify:
DefinitionSince ln x is not defined for negative values of x, you may frequently see ln|x|. The absolute value rule for ln is: 1 '
ln ln 'd d u
u u udx dx u u
When differentiating a logarithm, you may ignore any absolute value sign.
Try This
( ) ln sin(2 )f x xFind the derivative:
1cos(2 )(2) 2cot(2 )
sin(2 )x x
x
Don’t forget the chain rule
Try ThisFind the equation of the line tangent to:
lny x x at (1,1)
2 1y x
Lesson CloseThe natural logarithm is frequently used in Calculus. Be certain that you understand the properties of logarithms and know how to differentiate and integrate (next section) logarithmic functions.
Assignment
1. 324/15-29 Odd (Slides 1-14)
2. 324/31-63 Odd (Slides 15-36)