3.7 derivatives of logarithmic function mrs. macintyre ap.calculus
TRANSCRIPT
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 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 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