3.7 Derivatives of Logarithmic Function

Mrs. MacIntyreAp.Calculus

Rules for Derivatives of Logs

'( )u ude e u

dx

aaadx

d xx ln)(

dx

duaaa

dx

d uu ln)(

xx eedx

d)( x

xdx

d 1)(ln

1(ln )

d duu

dx u dx

)1

(ln

1)(log

xax

dx

da

'1 1(log ) ( )

lna

du u

dx a u

Remember' dyu

dx

Example 1uy ln

• Let u=• =

)1ln()( 3 xxf

1'

duy

u dx

23

1' (3 )

1y x

x

1

3'

3

2

x

xy

13 x23x

dx

du

Example 2)ln(sin)( xxf Let u=

uy ln'1

'y uu

1' (cos )

siny x

x

x

xy

sin

cos'

xy cot'

sin x' cosu x

Example 3• Use laws of log/Ins to

differentiate:

5

24

3

)23(

1

x

xxy

324

5

1

(3 2)

x xy

x

52/124/3 )23ln()1ln(ln(ln) xxxy

23 1(ln) ln ln( 1) 5ln(3 2)

4 2y x x x

)3)(23

1(5)2)(

1

1(

2

1)

1(

4

312

x

xxxdx

dy

y

23

15

14

312

xx

x

xdx

dy

y

)23

15

14

3(

12

xx

x

xy

dx

dy

y

23

15

14

3)(

)23(

)1(

25

24/3

xx

x

xx

xx

dx

dy

(ln) (ln)

d

dx

d

dx

Example 4( ) ( )xy x

xxy

ln( ) ( ln )y x x

xdx

dxx

dx

dx

dx

dy

ylnln

1

)2

1(ln)

1(

1 2/12/1 xxx

xdx

dy

y

x

x

xdx

dy

y 2

ln11

ln ln

d

dx

d

dx

ln both sides

Differentiate both sides

Product Rule:

1st derv 2nd + 2nd derv 1st

x

x

xdx

dy

y 2

ln

)2(

1)2(1

x

x

x

x

xdx

dy

y 2

ln2

2

ln

2

21

)2

ln2()

2

ln2(

x

xx

x

xy

dx

dy x

Example 4 Continued…..

Example 5• Let u=lnxxxf ln)(

1/ 2( ) (ln )f x x

2/1)( uxf

1/ 2 '1'( )

2f x u u

xxxf

1)(ln

2

1)(' 2/1

1'( )

2 lnf x

x x

xu

1'

Example 6• Let u =2+sinx

uy 10log

)sin2(log10 xy

dx

du

uy )

1(

10ln

1

xx

y cos)sin2

1(

10ln

1

cos'

ln10(2 sin )

xy

x

xu cos'

Remember

is the same

thing as

'duu

dx

Example 7)

2

1ln(

x

xy

ln( 1) ln 2y x x Use Rules for ln/logs to break up the problem into smaller easier parts.

12ln( 1) ln( 2)y x x

1ln( 1) ln( 2)

2y x x

Know do substitution for each term…. Lets call one term u and one term v….

1

2

u x

v x

Example 7 Continued…1ln( ) ln( )

2y u v 1

2

u x

v x

'

'

1

1

u

v

' ' '1 1 1

2y u v

u v

' 1 1 1(1) (1)

1 2 2y

x x

' 1 1

1 2( 2)y

x x

' 1 2( 2) 1 ( 1)

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

x xy

x x x x

' 2( 2) ( 1)

2( 2)( 1)

x xy

x x

' 2 4 1

2( 2)( 1)

x xy

x x

' 5

2( 2)( 1)

xy

x x

Example 82

3

8 1log

x

xy

)1(loglog32

1 288 xxy

)1(8ln

2

2

1

8ln

13

2

12x

x

x

)1(

23

8ln2

12x

x

x

Example 9x

xey

cos5

3

2cos

cos2cos

)5(

)sin(5ln5)3(533

x

xxxx xeex

xxx xxe

cos2

2cos

5

5lnsin35

3

xx xxe

cos

2

5

5lnsin33

Example 104

5

3

log xey

4

5ln

ln3xe

4

3

5ln

x

4

1

4

3

)5(ln

x

4

1

)5ln(

43

x

4

1

)5ln(4

3

x

Example 11xxexf ln)(

x

xxe xx lnln

)1(lnln xe xx

)1(ln xx x

Example 12 5sinln)( xxf

)sin(

)cos(55

54

x

xx

)cot(5 54 xx

Homework•Pg 250-251 #3-25 odd•27, 30, 33, 35, 38, 39, 40, 42

•Remember “you do homework for you not for me!”-good luck….