regra da cadeia- exercicios resolvidos

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Disciplina: Cálculo I Prof. Rogério Dias Dalla Riva 
Página 1 de 4 
Lista de Exercícios – A regra da cadeia 
 
 
1) Ache a derivada das funções abaixo. 
 
a) 3( ) (2 7)f x x= − 
 
 ( )' 2( ) 3 (2 7) 2 7df x x xdx= ⋅ − ⋅ − 
' 2( ) 3 (2 7) 2f x x= ⋅ − ⋅ 
' 2( ) 6(2 7)f x x= − 
 
b) 4( ) 3(9 4)g x x= − 
 
 
' 3( ) 4 3 (9 4) (9 4)dg x x x
dx
= ⋅ ⋅ − ⋅ − 
' 3( ) 12 (9 4) 9g x x= ⋅ − ⋅ 
' 3( ) 108(9 4)g x x= − 
 
c) 23( ) (9 2)f t t= + 
 
1
' 32( ) (9 2) (9 2)
3
df t t t
dt
−
= ⋅ + ⋅ + 
'
1
3
2( ) 9
3(9 2)
f t
t
= ⋅
+
 
'
1
3
6( )
(9 2)
f t
t
=
+
 
 
d) 3 33 4y x x= + 
 
 ( ) 1333 4y x x= + 
( ) ( )23' 3 31 3 4 3 43 dy x x x xdx
−
= ⋅ + ⋅ + 
( ) ( )
' 2
2
33
1 9 4
3 3 4
y x
x x
= ⋅ +
⋅ +
 
( )
2
'
233
9 4
3 3 4
xy
x x
+
=
+
 
 
e) 12 2( ) (25 )f x x −= + 
 
 
3
' 2 221( ) (25 ) (25 )
2
df x x x
dx
−
= − + ⋅ + 
Disciplina: Cálculo I Prof. Rogério Dias Dalla Riva 
Página 2 de 4 
3
' 2 21( ) (25 ) 2
2
f x x x
−
= − + ⋅ 
'
32 2
( )
(25 )
xf x
x
= −
+
 
 
2) Na função abaixo, determine uma equação da tangente ao gráfico 
de f no ponto (2, f(2)). 
 
a) 2( ) 4 7f x x= − 
 
( )122( ) 4 7f x x= − 
( ) ( )12' 2 21( ) 4 7 4 72 df x x xdx
−
= ⋅ − ⋅ − 
( )
'
1
22
1( ) 8
2 4 7
f x x
x
= ⋅
−
 
'
2
4( )
4 7
xf x
x
=
−
 
 
'
2
4 2 8 8(2)
394 2 7
m f ⋅= = = =
⋅ −
 
 
2(2) 4 2 7 9 3f = ⋅ − = = 
 
83 ( 2)
3
y x− = ⋅ − 
3 9 8 16y x− = − 
3 8 7y x= − 
 
8 7
3 3
y x= − 
 
3) Nos exercícios abaixo, ache a derivada da função. 
 
a) 
3 3
3( )
1
g x
x
=
−
 
 
 ( ) 133( ) 3 1g x x −= ⋅ − 
( ) ( )43' 3 31( ) 3 1 13 dg x x xdx
−
= − ⋅ ⋅ − ⋅ − 
( ) ( )43' 3 2( ) 1 3g x x x−= − − ⋅ 
( )
2
'
4
33
3( )
1
xg x
x
= −
−
 
Disciplina: Cálculo I Prof. Rogério Dias Dalla Riva 
Página 3 de 4 
( )
2
'
433
3( )
1
xg x
x
= −
−
 
 
b) 
2
2
3( )
2 1
tg t
t t
=
+ −
 
 
 
( )
2
1
22
3( )
2 1
tg t
t t
=
+ −
 
 
 
( ) ( ) ( )
( )
1 1
2 22 2 2 2
'
21
22
2 1 3 3 2 1
( )
2 1
d dt t t t t t
dt dtg t
t t
+ − ⋅ − ⋅ + −
=
 
+ − 
 
 
( ) ( )1 12 22 2 2 2
'
2
12 1 6 3 2 1 ( 2 1)
2( )
2 1
dt t t t t t t t
dtg t
t t
−
+ − ⋅ − ⋅ ⋅ + − ⋅ + −
=
+ −
 
( ) ( )21 12 22 2
'
2
32 1 6 2 1 (2 2)
2( )
2 1
tt t t t t t
g t
t t
−
+ − ⋅ − ⋅ + − ⋅ +
=
+ −
 
( ) 122
'
2 1
( )
t t
g t
−
+ −
=
( ) 22 32 1 6 2tt t t⋅ + − ⋅ − 2⋅
( ) 122
( 1)
2 1
t
t t
−
 
⋅ + 
 
+ − ( )322 2 1t t⋅ + −
 
( )
3 2 3 2
'
3
22
6 12 6 3 3( )
2 1
t t t t tg t
t t
+ − − −
=
+ −
 
( )
3 2
'
3
22
3 9 6( )
2 1
t t tg t
t t
+ −
=
+ −
 
( )
( )
2
'
32
3 3 2( )
2 1
t t t
g t
t t
+ −
=
+ −
 
 
c) 
2
2
6 5
1
xy
x
− 
=  
− 
 
 
 
'
2 2
6 5 6 52
1 1
x d xy
dxx x
− −   
= ⋅ ⋅   
− −   
 
( ) ( ) ( ) ( )
( )
2 2
'
2 22
1 6 5 6 5 16 52
1 1
d d
x x x x
x dx dxy
x x
− ⋅ − − − ⋅ −
− 
= ⋅ ⋅ 
− 
−
 
Disciplina: Cálculo I Prof. Rogério Dias Dalla Riva 
Página 4 de 4 
( ) ( ) ( ) ( )
( )
2
'
2 22
1 5 6 5 26 52
1 1
x x xxy
x x
− ⋅ − − − ⋅
− 
= ⋅ ⋅ 
− 
−
 
( )
2 2
'
2 22
6 5 5 5 12 102
1 1
x x x xy
x x
− − + − + 
= ⋅ ⋅ 
− 
−
 
( )
2
'
2 22
6 5 5 12 52
1 1
x x xy
x x
− − +
= ⋅ ⋅
−
−
 
( ) 2
'
2 3
2 6 5 (5 12 5)
( 1)
x x x
y
x
− − +
=
−