properties of logarithms log b (mn)= log b m + log b n ex: log 4 (15)= log 4 5 + log 4 3 log b...

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Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M log b N Ex: log 3 (50/2)= log 3 50 – log 3 2 log b M r = r log b M Ex: log 7 10 3 = 3 log 7 10

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Goals: Find derivatives of functions involving the natural logarithmic function. Use logarithmic differentiation to take derivatives of more complicated functions. Sect. 5-1 Derivatives of Natural Logarithmic Functions

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Page 1: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

Properties of Logarithms logb(MN)= logbM + logbN Ex: log4(15)= log45 + log43 logb(M/N)= logbM – logbN Ex: log3(50/2)= log350 – log32

logbMr = r logbM Ex: log7 103 = 3 log7 10

Page 2: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

5-1 The Natural Logarithmic Function 5-4 Exponential Functions 5-5 Bases other than e and Applications 5-3 Inverse Functions and their Derivatives 5-8 Inverse Trig Functions 7-7 Indeterminate Forms and L’Hopital’s Rule

Chapter 5 Derivatives of Transcendental Functions

Page 3: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

Goals:• Find derivatives of functions involving the natural logarithmic function.• Use logarithmic differentiation to take derivatives of more complicated functions.

Sect. 5-1 Derivatives of Natural Logarithmic Functions

Page 4: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

1lnd duudx u dx

Derivatives of Natural Logarithmic Function

ln ( )y f x

( )u f x

1lnd xdx x

0x

0u

1ln | |d xdx x

0x

Page 5: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

Example 1: Find the derivative of f(x)= xlnx.

Solution: This derivative will require the product rule.

1f (x) x lnx 1x

lnx1(x)f

Page 6: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

Example 2: Find the derivative of g(x)= lnx/x

Solution: This derivative will require the quotient rule.

2

1x lnx 1xg (x)

x

2xlnx1(x)g

Page 7: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

Example 3: Find the derivative of . )ln()( 1xxf 2

Solution: Using the chain rule for logarithmic functions.

( ) xf xx

2

21

Derivative of the inside, x²+1

The inside, x²+1

Page 8: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

YOU TRY…

1. Find the derivative of y = x²lnx .

3. Find [ln | cos |]d xdx

32. Find [ln( 4)]d xdx

Page 9: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

y’ = x² + (lnx)(2x)

= x + 2xlnx = x(1+2lnx)

x1

Example 1: Solution

Page 10: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

Example 4 Differentiate 632 21ln xxy

Solution: There are two ways to do this problem. One is easy and the other is more difficult.

The difficult way:

62 3 5 62 3 2 3

6 62 3 2 3

d x 1 x 2 x 1 6 x 2 3x x 2 2xdxyx 1 x 2 x 1 x 2

55 6 3 2 32 2 3 3

6 62 3 2 3

2x x 2 9x x 1 x 218x x 1 x 2 2x x 2y

x 1 x 2 x 1 x 2

3 3 3 4 2

2 72 3 2 32 3

2x 9x 9x x 2 2x 10x 9x 2 20x 18x 4xy

x 1 x 2 x 1 x 2x 1 x 2

Page 11: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

632 2x1xy ln

The easy way: First simplify the log using some of the expansion properties.

2x6ln1xln2xln1xln2x1xlny 32632632

2

2 3

6 3x2xyx 1 x 2

3 2 2

2 3 3 2

Now get a common denominator.

2x x 2 6 3x x 1y

x 1 x 2 x 2 x 1

4

3 2

2x +18xx 2 x 1

x x

4 24 18

4 2

2 3

20x 18x 4xx 1 x 2

Page 12: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

Example 5: Find the derivative of .

34ln

xxy

Expand to y ln x 4 ln x 3

3x1

4x1y'

Now get a common denominator and simplify.

x-3 x 4

x 4 x 3 x 4 x 3

7x 4 x 3

Page 13: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

Ex 6: Compute the derivative of .)3ln()( xxf

Note:

Therefore, we cannot use the properties of logs to bring the exponent down as a coefficient!

Instead, we must use the Chain Rule.

21

)3ln( x

21

)3ln( x 21

)3ln( x

x

xxf33)3ln(

21)(' 2

1

)3ln(21

xx

Page 14: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

YOU TRY…Find the derivative of each of the

following:

1.

2. f(x) = ln (x2 – 3)5

3. f(x) = [ln (x2 – 3)]5

4.

5.

1( ) ln2

xf xx

2 3

5 2

( 2)( ) ln1

xf xx x

3 1( ) ln5 2

xf xx

Page 15: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

ln (x2 – 3)5 = 5 ln (x2 – 3)

= ln (3x +1) – ln (5x-2)

3 1ln5 2

xx

Solution 1

Page 16: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

ddx

lnx 1

x 2

1x 1x 2

ddx

x 1

x 2

x 2x 1

x 2 1 (x 1) 12 (x 2) 1 2

x 2

x 2 1

2 (x 1)(x 1)(x 2)

x 5

2(x 1)(x 2)

Solution 4 Way 1

Page 17: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

If we first simplify the given function using the laws of logarithms, then the differentiation becomes easier:

12

1ln ln( 1) ln( 2)2

1 1 11 2 2

d x d x xdx dxx

x x

Solution 4 Way 2

Page 18: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

Solution 5

2 26 1 2

2 5( 1)y x xy xx x

dxdy

2 3

5 2

( 2)ln ln1

xyx x

2 21ln 3ln( 2) (ln ln 1 )5

y x x x

Page 19: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

ClosureExplain how to differentiate the following function both the hard way and the easy way.

2 3

5 2

( 2)( ) ln1

xf xx x

Page 20: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

Taking this derivative would involve the Power, Product, Quotient, and Chain (twice) Rules.

Instead we can use a technique called logarithmic differentiation to simplify the process.

3/4 2

5Differentiate: (

13 2)

x xyx

LOGARITHMIC DIFFERENTIATION

Page 21: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

Take the “ln” of both sides and use properties of logarithms to expand.

Differentiate implicitly with respect to x.

2

1 3 1 1 2 354 2 3 21

dy xy dx x xx

23 14 2ln ln ln( 1) 5ln(3 2)y x x x

3/4 2

5Differentiate: (

13 2)

x xyx

Page 22: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

Solve for dy / dx.

Since we have an explicit expression for y, we can substitute and write:

3 / 4 2

5 2

1 3 154 3 2(3 2) 1

dy x x xdx x xx x

2

3 154 1 3 2

dy xydx x x x

Page 23: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

1. Take natural logarithms of both sides of an equation y = f(x) and use the Laws of Logarithms to simplify.

2. Differentiate implicitly with respect to x.

3. Solve the resulting equation for y’.4. Replace y with f(x).

STEPS IN LOGARITHMIC DIFFERENTIATION

Page 24: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

Example 1: Differentiate . ))(( 1x1xxy 2

Solution: This problem could be easily done by multiplying the expression out.

))((lnln 1x1xxy 2

ln ln ln( ) ln( )y x x x 21 1y xy x x x

21 1 2

1 1

Page 25: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

1x2x

1x1

x1

yy

2

1x

x21x

1x1yy 2

))(( 1x1xxy 2

22

1 1 2xy x x 1 (x 1)x x 1 x 1

Page 26: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

12x3x4xy

2x2xxx1xxxy

1xx2x1)x(x1)(x1xy

1x1)(x1xx2x

1x1)(x1xx

x1)(x1xx

23

23323

22

2

222

y

Continue to simplify…

Page 27: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

Let’s double check to make sure that derivative is correct by multiplying out the original and then taking the derivative.

1x2x3x4y

xxxx1xxxy1x1xxy

23

23422

2

)(

))((

Remember this problem was to practice the technique. You would not use it on something this simple.

Page 28: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

Chain Rule and Product Rule

Ex 2. 2Find '( ) given ( ) ( 3) 5 4.h x h x x x 1

2 2( 3)(5 4)x x 1 1

2 2 21'( ) ( 3) (5 4) 5 (5 4) 22

h x x x x x

25( 3) 2 (5 4)

2 (5 4)x x x

x

This time use logarithmic differentiation to solve.

Page 29: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

Chain Rule and Quotient Rule 72 1Find '( ) given ( ) .

3 5xG x G xx

6

2

3 5 2 2 1 32 1( ) 73 5 3 5

x xxG xx x

66

2 8

91 2 12 1 13( ) 73 5 3 5 3 5

xxG xx x x

Ex 3.

This time use logarithmic differentiation to solve.

Page 30: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

3 24

11. Find 6 2 5

ddr r r

You Try…

3 2 42. Find given ( 3 ) (2 1) .dy y x x xdx

3. Find dy/dx if . 2 1/2( 1)( 3)

1x xy

x

Page 31: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

1 2 1xdy x xdx

1ln ln 2 xy x

ln 1 ln 2y x x

1 11 ln 22

dy x xy dx x

1 ln 22

dy xy xdx x

1 12 ln 22

xdy xx xdx x

Logarithmic differentiation is also used when the variable is in the base and the exponent.

.)2( ateDifferenti :4Ex 1 xxy

)()( xgxfy

Page 32: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

Example 5: Differentiate .

ln ln lnxy x x x

xy x

' 1 1(ln )2

y x xy x x

1 ln 2 ln'2 2

xx xy y xx x x

Page 33: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

Ex 6: Find an equation of the tangent line

to the graph of at .,12

2sin

xy x

Page 34: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

1. y = xx

You Try…

Differentiate each of the following functions.

2. y = (sinx)lnx

xxdxdy x ln1

Page 35: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

ln y = ln xx

ln y = x ln x

xx

xdxdy

yln11

xydxdy ln1

xxdxdy x ln1

y = xx

Page 36: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log
Page 37: Properties of Logarithms log b (MN)= log b M + log b N Ex: log 4 (15)= log 4 5 + log 4 3 log b (M/N)= log b M – log b N Ex: log 3 (50/2)= log 3 50 – log

ClosureExplain the steps for the different methods used to differentiate the following functions.1. y = x 4

2. y = xx