3.9: DERIVATIVES OF
EXPONENTIAL & LOGARITHMIC
FUNCTIONS
HOMEWORK: 3-42 MULT OF 3, 43-46
Warm-up:
Using the definition of a derivative, find the derivative of
d
Xth
- . (eX) ~ \Im e
dx
h~O
-= IHl\
h~O
- tX
h
e~· eh -
tx
h
1
Derivative of eX
d
du
dx
dx
-ell = e ll -
Examples:
(a)
d)
d~
(t 3>.
t
I)
: e3x t I " 3
-- 3e 3x t I
y = e 3x + 1
(b) Y = e s inx
6 ) e SIr\)<
d
dX (e Slnx)
=e
S1nx
"cosx
2
Derivative of aX
-
d
du
-(aU) = aU lna­
dx
dx
(a is a constant)
Examples:
(a)
y = 3cotx
(b) y =
S2X-2
b)
! (5
=
ZX- 2 )
S2x- 2
\\\5 ° 2
3
Derivative of In x
d
1 du
-lnu = - ­
dx
u dx
Examples:
(a)
J)
d
~ (It) (1x-l)
- \
1A -I
;;
)
·1
y = In(4x - 1)
(b) y = In (cos x)
b) d (
.
dx in I(Os).) )
­-
--
loS X
,;;
- SlhX
i
4- x-I
--
-Sln)(
(OSX
==
-tanx
4
Derivative of loga x
d
dx loga U =
du
U In a dx
1
Examples:
(a)
y = [oglOe X
(b) y = log(3x
d~ { lo~ (3X i I) )
b)
-
:;
+ 1)
I
(c) Y = logs..Jx
c) ~~ \C9s-rx
~:: \~ S X'/1
-(3Xt ,) Ih ID- · 3
~:: ~ . \O~ 5 ~
3
-(3X1"I)
-In 10
i'=l~l ' l
2
~I ~
X \r{)
___
, _
2x IV) 5
~ I::
I
• I
--IX \h5
2{X
::­
2x InS
5
Rules of log's & In's that may be useful•••. •
e 1nx
=x
6
Extra Practice Problems:
2x
3. y = e3
2 . y = eX Inx
1. rf:ODU(T KULE
U-=-X
1
v-:.e x
\J I ~
2><
Vi ~
eX
2. rRoDuC1 k'u lE.
UI = eX
\j=-e
).
v -: Inx
V'-;:_I
- ~2e x t
2xe x
-- 'f..e" (x t2) ::.
~
X.
1
e),lnx
'"
, e (~ t Inx)
1
e-;)\
.2 3
=z
3
~
'JI ~"
7
Extra Practice Problems:
10
5. Y = In(4x)
4. Y = In-;-
4- . \D \0 - \nx
! ( \\\10 - Inx)
dX
~
0 -~
5 . \n
id~ (\n~ i \n x)
~
A
~ - \
X
(~x)
-=
01 .l.
X
I
~. ~~
9X
idx (f)
-;g~\n~ ' 1
=
9A\fI ~
X
8
Harder Practice Problems:
10. y=xX
.
~
X
\0. ~::
1n~ <1\ tnX
ldl;; ~~ .~ t\Y\x . \
~ (:\x
'A
-
I dlA
_J
J dx
~~
0
~
\
-t \
3
2
'2 I cl~ _ '2'J.
+ ---­
3- ~ dx - X~t I 2(3Xt1) - 5(2x-4)
nx
(Itlhxl~
9