مشتقة الدوال الأسية واللوغاريتمية · :1 3 Derivative of the...
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: 1 3 Derivative of the Exponential and Logarithmic functions ﻣﺮ ا ﺟﻌﺔ ﻟﻘﻮاﻋ ﺪ اﻟﺪاﻟﺔ اﻷﺳﯿﺔ: (1 1 3 ) Inverse-function rule (2 1 3 ) 2) , 0 m m n n x x x x 1 3) n x n x 0 4) 1 x 1) m n m n x x x 1 5) n n x x 6) n m mn x x 7) m m m xy x y 8) 0, 0, 0 : If u b a then u u a b a b 9) 1, : If a then u v a a u v 0 1, : And If a then u v a a u v Rules of exponents log ; log ; 0, 1, 0 a x x a a x a x a a x
Transcript of مشتقة الدوال الأسية واللوغاريتمية · :1 3 Derivative of the...
:1 3
Derivative of the Exponential and Logarithmic functions
:الدالة األسیة دلقواع جعةامر (1 1 3)
Inverse-function rule ( 2 1 3)
2) , 0m
m n
n
xx x
x
13) nxnx
04) 1x
1) m n m nx x x
1
5) n nx x
6)nmmnx x
7) m m mxy x y
8) 0 , 0 , 0 :If u b a then u ua b a b
9) 1, :If a then u va a u v
0 1, :And If a then u va a u v
Rules of exponents
log ; log ; 0 , 1 , 0a x xaa x a x a a x
Product rule : log log loga a axy x y
Quotient rule : log log loga a a
xx y
y
Power rule : log logya ax y x
lnChange of base formula : log
lna
xx
a
: 3 3eSome Definitions of e
(1 2 3) 1
1) lim 1n
xe
n
2) ( 2 2 3) 0
1lim 1
h
x
e
h
e
( 3 2 3) 0
13)
!n
en
(e) irrational number
1lim 1 .
n
ne
n
2 71828
( 3 1 3) DERIVATIVES OF EXPONENTIAL FUNCTIONS
: ( 4 1 3) 1) ( ) ( )x xf x e f x e
0 0
( ) ( )( ) lim lim
x h x
h h
f x h f x e ef x
h h
0 0
1lim lim ( )
x h x hx
h h
e e e ee
h h
0 0
1lim lim 1
hx x x
h h
ee e e
h
: 5 1 3 ( ) ( )2) ( ) ( ) ( )u x u xf x e f x u x e
y = u ( x );f (x ) = g yg e
Chain Rule
( )( ) ( )y u xdg dg dye u x u x e
dx dy dx
133 2( ) x xf x e
33 2 3( ) (3 2 )x xf x e x x
32 3 2(2 3 ) x xx e
sin(2 1)( )f e
sin(2 1)( ) sin(2 1)f e sin(2 1)2 cos (2 1) e
: 3 4 3 lny x1dy
dx x
xln yy x x e
1ln
ln 11 y y x x
dy dye e e e
dx dx x
: 4 4 3 lny uy , ux1dy du
dx u dx
: 6 1 3 ln | |y x1dy
dx x
x > 0| x | = x lny x1dy
dx x
x < 0| x | = - x ln ( )y x
1 1( 1)
dy
dx x x
: 7 1 3
:8 1 3
dy
dx
2
3ln( )
1
xy
x
21ln(3 ) ln( 1)
2y x x
1 1
2 3
dy
dx 3
x
2 2
1 1 1 22
1 2 1
xx
x x x
5sin 3 cos 2 tany x x x
ln ln 5 ln sin 3 ln cos 2 ln tany x x x
(ln ) (ln 5 ln sin 3 ln cos 2 ln tan )d d
y x x xdx dx
21 3cos3 2sin 2 1 sec0
sin 3 cos 2 tan
dy x x x
y dx x x x x
21(5sin 3 cos 2 tan )( 3cot 3 2 tan 2 sec cot )
dyx x x x x x x
dx x
1 , 0xy x x x
ln lny x x
(ln ) ( ln )d d
y x xdx dx
1 1ln (1 ln )xdy dy
x x x xy dx x dx
2ln|3 2 5|x xy e
| |ln ue u2ln | 3 2 5 |y x x
2
6 2
3 2 5
dy x
dx x x
2( ) 4 12x xf x e e x x ( ) 4f x
2( ) 2 4 12x xf x e e
22 4 12 4x xe e 2
2 22 4 16 0 2 8 0x x x xe e e e
( 4) ( 2) 0 2
x x
غ�����������������یرممكن
xe e ore
4 ln 4xe x
( 1 3)
