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Maths' formula sheet
Transcript of Τυπολογιο οριων & παραγώγων
-
, /
: 0
lim ( )x x
f x
0, 0: 0x x
( )f x
) ) )
( ) ( ) ( , )f x f x
)
0x x 0 0( , )x x x
1lim 3 2 1x x
3 2 1x 3 1x
13
x , 3
.
0 1 1lim ( )x x f x &
0 2 2lim ( )x x f x :
1. 0 1 2 1 2
lim ( ) ( )x x f x f x n- 0 0 0
lim sin ln limsin lim lnx x x x x x
x x x x
2. 0 1 2 1 2
lim ( ) ( )x x f x f x n- 0 0 0
lim tan lim tan limx xx x x x x x
x e x e
3.
0 0 01 2 1 2 1 2lim[ ( ) ( )] lim ( ) lim ( )
x x x x x xf x f x f x f x
n- 0 0 0
3 2 2 2lim cos lim lim cosx x x x x x
x x x x
3().0 1 1
lim ( )x x kf x k , k R 0 0
lim 3cos 3 lim cosx x x x
x x
3(). 0 0
lim ( ) lim ( )n
n
x x x xf x f x , n N
( ) 0f x
0
lim ( ) 0x x
f x 0 0
5 5lim ( lim )x x x x
x x
4. 0
1 1
2 2
( )lim
( )x x
f x
f x
2 0
4
4
4
2lim sin sin
sin 4 2lim 1cos lim cos 2cos
4 2
x
x
x
xx
x x
5. 0 0
lim ( ) lim ( )n nx x x x
f x f x , *n N n=
3 2 23
5 5lim 3 5 lim( 3 5)x x
x x x x
6. 00
lim ( )( )lim x x
f xf x
x xa a , 0a
22
0lim 3
3
0lim x
x xx x
xe e
: lim ( )x
f x
0, 0 : ( )M x M f x
) . ) .
1lim 0x x
, 01 1 1
0x
x Mx x
x,
22 2
22
2
1 1(1 5 6 )
5 6 1lim lim
1 13 2 2 3(3 2 2 )
x x
xx x x x
x xx
x x
lim
1x
x
x x
( 1 )lim
( 1 )( 1 )x
x x x
x x x x
x ()
2lim 1x
x x ax2
1 1lim 1x
x ax x
x
2lim 1x
x x ax = 2lim 1x
x x x ax x
0
lim ( )x x
f x
0 0
lim ( ) lim ( )x x x x
f x f x
, , .
3 1, 1
1
2 1 1( ) , 0 1
1 1
11, 0
1
xx
x
x x xf x x
x x
xx
x
: 0
0lim ( ) ( )x x
f x f x
f x0.
-, , , , .
0
0lim sinsinx x
xx , 0
0lim ln lnx x
x x ,
0
0limx x
xxa a
-
, /
:
0
0
0
( ) ( )limx x
f x f x
x x=
0'( )f x = 0( )df
xdx
,
f x0.
0'( )f x
( )f x x
0 0
0 00
0 0
( ) ( )'( ) lim lim 1
x x x x
f x f x x xf x
x x x x
f , g
1. [ ( ) ( )]' '( ) '( )f x g x f x g x n-
2. ( )'( ) '( ) ( ) ( ) '( )f g x f x g x f x g x n-
2(a) ( )'( ) '( )k f x k f x , k R
2(b) 1( ( )) ' ( ) '( )n nf x n f x f x , n R 1( ) 'n nx n x
3. 2
'( ) ( ) ( ) '( )( ) ( )
( )
f f x g x f x g xx
g g x
2 2
sin (sin )'cos sin (cos )' 1(tan )'
cos cos cos
x x x x xx
x x x
4. (sin ) ' cosx x (cos ) ' sinx x
2
cos 1(cot ) '
sin sin
xx
x x
5. ( ) '( ( )) '( )f g x f g x g x
2 2 2 2sin( 2 ) cos( 2 )( 2 ) cos( 2 )(2 2)x x x x x x x x x
6. ( ) ( ) '( )f x f xe e f x x xe e
7. 1
1
'( )( ) ( )
( )
kk
kk
f xf x f x
k f x
. ( )
'( )( )
2 ( )
f xf x
f x,
1
2x
x
8. '( )
ln ( )( )
f xf x
f x
1(ln )'x
x,
1(log )' loga ax e
x, 0a
9. 1 00
1( ( ))
'( )f f x
f x
( )
10. 12
'( )sin ( ( ))
1 ( )
f xf x
f x
12
1sin ( )
1x
x
11. 12
'( )cos ( ( ))
1 ( )
f xf x
f x
12
1cos ( )
1x
x
12. 12
'( )tan ( ( ))
1 ( )
f xf x
f x
12
1tan
1x
x
13. 12
'( )cot ( ( ))
1 ( )
f xf x
f x
12
1cot
1x
x
14. sinh cosh2
x xe ex x
Sinhx=
15. cosh sinh2
x xe ex x
Coshx=
16. (ln 1)x xx x x
lnxx knx x xx e e
( ) ( ) 1( '( )ln ( ) )f x f xx x f x x f xx
,
f .
