Aim: Differentiating Natural Log Function Course: Calculus Do Now: Aim: How do we differentiate the natural logarithmic function? Power Rule

Aim: Differentiating Natural Log Function Course: Calculus

Do Now:

Aim: How do we differentiate the natural logarithmic function?

5

1

1Evaluate: dx

x

1

, 11

nn x

x dx C nn

Power Rule


Exponential Equation

y = bx

Inverse Exponential Function

y = logbxLogarithmic Equation

Inverse of Exponential Equation

x = by

Exponential example

y = 2x

y = log2xLogarithmic example

Inverse of Exponential example

x = 2y

logb x = y if and only if by = xThe expression logb x is read as the “log base b of x”. The function f(x) = logb x is the logarithmic function with base b.

Logarithm = Exponent

y = bx

“x is the logarithm of y”

y = logbx

“y is the logarithm of x


Natural Logarithmic Function

f(x) = logex = ln x, x > 01. ln 1 = 0

2. ln e = 1

3. ln ex = x

because e0 = 1

because e1 = e

because ex = ex

4

2

-2

-4

-5 5

u x = lnx

4

2

-2

-4

-5 5

v x = ex

The logarithmic function with base eis called the natural

log function.

4. e ln x x inverse property5. If ln x = ln y, then x = y

1lim 1

x

xe

x


Properties of Natural Log

1. The domain is (0, ) and the range is (-, ).

2. The function is continuous, increasing, and one-to-one.

3. The graph is concave down.

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

• ln(1) = 0

• ln(ab) = ln a + ln b

• ln(an) = n ln a

• ln (a/b) = ln a – ln b

4

2

-2

-4

-5 5

u x = lnx


Using Properties of Natural Logarithms

2. ln e2

3. ln e0

Rewrite each expression:

e

11. ln

ln ex = x because ex = ex

= -1

= 2

= 0

4. 2ln e = 2 ln e = 1 because e1 = e

e 1ln

ln ex = x

ln ex = x


Model Problems

Use natural logarithms to evaluate log4 30

log b M log c M

log c b

`

log4 30ln30

ln4

3.401198...

1.38629...2.4538....

Given ln 2 0.693, ln 3 1.099, and ln 7 1.946, use the properties of logs to approximate a) ln 6 b) ln 7/27

ln 6

ln 7/27

= ln (2 • 3) = ln 2 + ln 3 0.693 + 1.099 1.792= ln 7 – ln 27 = ln 7 – 3 ln 3 1.946 – 3(1.099) -1.351


Model Problems

Use properties of logarithms to rewrite

ln3x 5

7 = ln(3x – 5)1/2 – ln 7

= 1/2 ln(3x – 5) – ln 7

22

23

3ln

1

x

x x

2 2 23

1 32 2

1 32 2

2 2

ln( 3) ln 1

2ln( 3) ln ln 1

2ln( 3) ln ln 1

12ln( 3) ln ln 1

3

x x x

x x x

x x x

x x x


Power Rule – the exception

3 32 2

2 21 1

0

1

1 2 12

3 3

2 2

??? ???

x

x

x

x

x

x xD x x dx C

x xD x x dx C

D x x dx x C

dxD x C

xdx

D x x x Cx



1

, 11

nn x

x dx C nn

Power Rule

5

1

1Evaluate: dx

x5 1

1x dx

5 51 1 0

1 1

!!!!0 0

x x

don’t work!



1

, 11

nn x

x dx C nn

Power Rule

no antiderivative for f(x) = 1/x

1

1ln , 0

xx dt x

t

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

numbers.

Definition of the Natural Logarithmic Function

2nd Fundamental Theorem of Calculus

( )x

a

d dFf t dt f x

dx dx

accumulation function


Definition of the Natural Log Function

1

1ln , 0

xx dt x

t

4

3.5

3

2.5

2

1.5

1

0.5

1 2 3 4

f x = 1

x

4

3.5

3

2.5

2

1.5

1

0.5

1 2 3 4

f x = 1

x

ln x is positive when x > 1

1

1If 1, 0

xx dt

t

x x

1

1If 0 < 1, 0

xx dt

t

11

ln x is negative when x < 1

ln(1) = 01

1

10dt

t

4

2

-2

-4

-5 5

u x = lnx

The natural log function measures the area under the curve f(x) = 1/x between 1 and x.


e

1

1ln

xx dt

t

1

1ln 1

ee dt

t

What is the value of x?

4

3.5

3

2.5

2

1.5

1

0.5

1 2 3 4

f x = 1

x

e1

1

1Area = 1

edt

t

ln 1e 1log 1e e e e

ln ne

ln

1

n e

n

n

1

e

x

4

2

-2

-4

-5 5

u x = lnx

(e, 1)


The Derivative of the Natural Log Function

Let be a differentiable function of .

1ln , 0

u x

dx x

dx x

Chain Rule

1 'ln , 0

d du uu u

dx u dx u

1

1 1ln 0

xd ddt x x

dx t x dx

2nd Fundamental Theorem of Calculus

( )x

a

d dFf t dt f x

dx dx

1

1ln , 0

xx dt x

t


Model Problems

1. ln 2d

xdx

2u x

' 2 1ln 2

2

d ux

dx u x x

u’ = 2

2. lnd

xdx

u x

1 21 2

1 1 1 1 1ln '

2 2

du x

dx u u x x

1 21'

2u x


Model Problems

2ln 1d

xdx

2

' 2

1

u x

u x

lnd

x xdx

ln lnd d

x x x xdx dx

1ln 1 1lnx x x

x

2 1u x u’ = 2x

3ln

dx

dx 2

3 ln lnd

x xdx

2 13 ln x

x


Rewrite Before Differentiating

Differentiate ( ) ln 1f x x

rewrite 1 2 ( ) ln 1f x x

1 ln 1

2x

1 1 1

'( )2 1 2 1

f xx x

1 1 ln 1 ln 1

2 2

d dx x

dx dx

1u x u’ = 1 1 'ln

d du uu

dx u dx u


Model Problem

22

3

1Differentiate ( ) ln

2 1

x xf x

x

1 'ln

d du uu

dx u dx u

rewrite 2 31ln 2ln 1 2 1

2x x x

'( )f x

u = x2 + 1 u = 2x3 – 1 u’ = 2x u’ = 6x2

2

2 3

1 4 3

1 2 1

x x

x x x

1

x 2

22

1

x

x

2

3

1 6

2 2 1

x

x


Logarithmic Differentiation

2

2

2Differentiate

1

xy

x

14

12

10

8

6

4

2

-2

5 10 15

g x = x-2 2

x2+1 0.5

y is always positive therefore ln y is defined

take ln of both sides

2

2

2ln ln

1

xy

x

Log properties 21ln 2ln 2 ln 1

2y x x

2

' 1 1 22

2 2 1

y x

y x x

Differentiate

Applying the laws of logs to simplify functions that include quotients, products and/or powers can simplify differentiation.


Using Log Derivative

2

2

2Differentiate

1

xy

x

2

' 1 1 22

2 2 1

y x

y x x 2

2

2 1

x

x x

Solve for y’,

Substitute for y

& Simplify

2

2'

2 1

xy y

x x

2 2

22

2 2 2

2 11

x x x

x xx

2

3 22

2 2 2

1

x x x

x


Derivative Involving Absolute ValueIf is a differentiable function of

such that 0, then

'ln

u x

u

d uu

dx u

Find the derivative of f(x) = ln|cosx|

'ln cos

sin

costan

d ux

dx ux

xx

u = cosx u’ = -sinx


Model Problem

2Find the relative extrema of ln 2 3y x x u = x2 + 2x + 3

u’ = 2x + 22

' 2 2'

2 3

u xy

u x x

2

2 20 1

2 3

xx

x x

1st Derivative Test

f(-1) = ln[-12 + 2(-1) + 3] = ln 2

Relative Extrema – (-1, ln2)

Evaluate critical point

Minimum

2

22

2 4 2''

2 3

at 1 positive result

concave up

x xy

x x

x

2nd Derivative Test

6

4

2

-5 5

g x = ln x2+2x+3


Do Now:

Aim: How do we differentiate the natural logarithmic function?

2

lnFind the derivative for ( )

tg t

t

Documents

Aim: Differentiating Natural Log Function Course: Calculus Do Now: Aim: How do we differentiate the natural logarithmic function? Power Rule