4ª_Lista_Exercícios_Cálculo_I_

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Engenharia de Produção 
4ª LISTA DE EXERCÍCIOS – Cálculo I (2012/1) PROF. GEOVANE OLIVEIRA 
Derivadas 
 
Exercício 1: 
1) 763 23)( ++= xxexf 23 6 7'( ) 3 (6 6)x xf x e x+ += ⋅ + 
2) xexf −= 3
3
1)( 31'( )
3
xf x e −= − 
3) xxxxf 643 234)( ++= 3 23 4 6 2'( ) 4 ln 4 (9 8 6)x x xf x x x+ += ⋅ ⋅ + + 
4) xexxxf 332 2)167()( −+−+= 2 2 3'( ) 3(7 6 1) (14 6) 6 xf x x x x e−= + − ⋅ + − 
5) )5()( 22 ttetf
t
+= 22 2
1
'( ) ( 5 ) (2 5)
2
t t
f t e t t e t= + + + 
6)
t
e
tf
t 1)(
2
+
= 
( ) ( )2 2
2
2 1
'( )
t te t t e
f t
t
− +
= 
7)
1
1)(
+
−
=
x
x
e
e
xf ( )
1
2
2
1 1 ( 1) ( 1)( )
2 1 1
x x x x x
x
x
e e e e ef x
e e
−  
 − + − − 
= ⋅   +  + 
 
8) 23 32)( xxexf −= ( )3 2 12 3 3 2 221'( ) 2 3 (6 6 )2x xf x e x x x x
−
−
= ⋅ − − 
9) )423sen()( 2 +−= xxxf 2'( ) cos(3 2 4) (6 2)f x x x x= − + ⋅ − 
10) 2( ) cos( 4 2 4)f x x x= − + + 2'( ) sen( 4 2 4) ( 8 2)f x x x x= − − + + ⋅ − + 
11) xexf x 3cos.)( 2= 2 2'( ) 2 cos3 3sen3x xf x e x e x= − ⋅ 
 
12) 42223 )162()35()( +−⋅+−= xxxxxxf 
3 2 2 2 4 3 2 2 2 3
'( ) 2(5 3 )(15 6 1) (2 6 1) (5 3 ) 4(2 6 1) (4 6)f x x x x x x x x x x x x x x= − + − + ⋅ − + + − + ⋅ − + −
 
13) ( ) ( )22323 35434)( xxxxxxf −⋅−+= 
( ) ( ) ( ) ( ) ( )2 2 32 3 2 2 3 2 3 2'( ) 36 18 12 4 3 4 5 3 4 3 4 100 90 18f x x x x x x x x x x x x x x= + − + − ⋅ − + + − ⋅ − +
 
14) ( ) ( )( )3232 1923735)( +−+⋅+−= xxxxxxf 
( ) ( )( ) ( ) ( )( )22 3 2 3 2 2 2( ) 3 5 3 7 3 2 9 1 (10 3) 3 2 9 1 5 3 7 9 4 9f x x x x x x x x x x x x x x = − + ⋅ + − + ⋅ − + − + + − + + − 
 
U N I V E R S I D A D E U N I G R A N R I O 
PROF. JOSÉ DE SOUZA HERDY 
15) ( )45 3 2( ) 2 3 2 4 3f x sen x x x x−= + − + + 
( ) ( )4 35 3 2 5 3 2 6 2'( ) cos 2 3 2 4 3 .4 2 3 2 4 3 ( 10 9 4 4)f x x x x x x x x x x x x− − −= + − + + + − + + ⋅ − + − +
 
16) 
3
2
2( ) ln
1
x xf x
x
 +
=  
− 
 
4 2
5 3
4 14 2
'( )
4 2 2
x xf x
x x x
− −
=
− −
 
 
17) ( )134sen)( 2 ++= xxxf ( ) ( ) 12 2 21'( ) cos 4 3 1 4 3 (8 3)2f x x x x x x−= + + ⋅ + + 
 
18) ( ) ( )( )1223cos)( 234 −⋅−= xxxxf 
 ( ) ( )( ) ( )4 3 2 5 4 3 2'( ) sen 3 2 2 1 36 20 12 6f x x x x x x x x= − − ⋅ − ⋅ − − + 
 
19) ( )134)( 2 +−= xxtgxf ( )2 2'( ) sec 4 3 1 (8 3)f x x x x= − + ⋅ − 
20) 
62
3 2
3 1( )
5
x xf x tg
x x x
 − +
=  
− + 
 
( )( )6 5 3 2 2 22 22
3 2 3 2 3 2 2
(2 3)( 5 ) 3 1 3 10 13 1 3 1
'( ) sec .6
5 5 ( 5 )
x x x x x x x xx x x xf x
x x x x x x x x x
 
− − + + − + − +   
− + − +
 = ⋅     
− + − + − +     
 
21) 








−
+
=
1
1
sec)(
x
x
e
e
xf 
 
1 1
'( ) sec tg
1 1
x x
x x
e ef x
e e
   + +
= ⋅   
   
− −    ( )
1
2
2
1 1 ( 1) ( 1)
2 1 1
x x x x x
x
x
e e e e e
e e
−  
 + − − + 
⋅ ⋅   
−  − 