Lista 4 de CDI derivadas2 A regra da cadeia

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UNIFESP DE OSASCO 
Lista 4 de CDI I –Prof. Rosângela 
Regras de derivação. A regra da cadeia 
FORMULÁRIO 
Tabela de derivadas 
Sejam u= u(x) e v=v(x) funções deriváveis e seja c uma constante. 
1) (c)' = 0 
2) (cv)'= cv' 
3) (u+v)' = u'+v' 
4) (uv)' = u'v + v'u 
5) 
2
''
'
v
uvvu
v
u 






 
6) 
2
´
'
1
v
v
v







 
7) (xn)' = nxn-1 (un)'= nun-1u' 
8) (senx)´= cos x (senu)´= u´cosu 
9) (cosx)´= -senx (cosu)´= -u´senu 
10) (tgx)´= sec2 x (tgu)´=u´. sec2 u 
11) (cotgx)´= - cosec2 x (cotgu)´= -u´. cosec2 u 
12) (secx)´= secx .tgx (secu)´= u´secu .tgu 
13) (cosecx)´= - cosecx .cotgx (cosecu)´= -u´cosecu .cotgu 
14) (ex)´=ex (eu)´=u´.eu 
15) (lnx)´=1/x (lnu)´=u´/u 
 
 EXERCÍCIOS 
 
1) Usando as fórmulas de derivação, derive e simplifique. 
a) 
5
1
)(
x
xf 
 
Resp: 
6
5
)´(
x
xf


 
b) 
4
1
)(
x
xf 
 
Resp: 
5
4
)´(
x
xf


 
c) 
nx
xf
1
)( 
 
Resp: 
1
)´(



nx
n
xf
 
d) 
8
6
)(
x
xf 
 
Resp: 
9
48
)´(
x
xf


 
e) 
44
3
)(
x
xf 
 
Resp: 
5
3
)´(
x
xf


 
f) 
34 3
2
4
3
)(
xx
xf 
 
Resp: 
5 4
3 2
(´ )f x
x x
  
 
g) 
x
xf
1
)( 
 
Resp: 
32
1
)´(
x
xf


 
h) 
3
1
)(
x
xf 
 
Resp: 
3 43
1
)´(
x
xf


 
i) 
4
1
)(
x
xf 
 
Resp: 
4 54
1
)´(
x
xf


 
j) 
n x
xf
1
)( 
 
Resp: 
n nxn
xf
1
1
)´(



 
k) 
x
xf
2
)( 
 
Resp: 
3
1
)´(
x
xf


 
l) 
3
3
)(
x
xf 
 
Resp: 
3 4
1
)´(
x
xf


 
m) 
4
4
)(
x
xf 
 
Resp: 
4 5
1
)´(
x
xf


 
n) 
65
3
)(
x
xf 
 
Resp: 
6 710
1
)´(
x
xf


 
o) 
xx
xf
4
32
)(
3

 
Resp: 
33 4 8
3
3
2
)´(
xx
xf 
 
p) 
xxxx
xf
1234
)(
234

 
Resp: 







2345
14916
)´(
xxxx
xf
 
q) 
xxxx
xf
2345
)(
345

 
Resp:









33 44 55 6
1111
)´(
xxxx
xf
 
r) 
xxxxxf  345)(
 
Resp: 
xxxx
xf
2
1
3
1
4
1
5
1
)´(
3 24 35 4

 
s) 
xxxf 3)( 
 
Resp: 
2
7
)´(
5x
xf 
 
 
t) 
4
3
)(
x
xx
xf 
 
u) Resp:
4
13
)´(
4 9x
xf 
 
 
2) Nova fórmula de derivação (a regra da cadeia): 
 Se u = u(x) é uma função derivável então 
 [(u(x))
n
] = n(u(x))
 n-1
u´(x) 
 ou 
 [u
 n
 ]´= nu
n-1
u´ 
Usando as fórmulas de derivação da tabela e a nova fórmula de derivação acima, derive e simplifique. 
 
a) 
4)13(
1
)(


x
xf
 
Resp: 
5)13(
12
)´(



x
xf
 
b) 
3)13(
1
)(


x
xf
 
Resp: 
4)13(
9
)´(



x
xf
 
c) 
13)(  xxf
 
Resp: 
132
3
)´(


x
xf
 
d) 
3 13)(  xxf
 
Resp: 
3 2)13(
1
)´(


x
xf
 
e) 
12)(  xxf
 
Resp: 
12
1
)´(


x
xf
 
f) 
42 )1(
1
)(


x
xf
 
Resp: 
52 )1(
8
)´(



x
x
xf
 
g) 
3)31(
1
)(
x
xf


 
Resp: 
4)31(
9
)´(
x
xf


 
h) 
14)(  xxf
 
Resp: 
14
2
)´(


x
xf
 
i) 
3 3 3)( xxxf 
 
Resp: 
3 23
2
)3(
1
)´(
xx
x
xf



 
j) 
xxxf 4)( 4 
 
Resp: 
xx
x
xf
4
22
)´(
4
3



 
k) 
61)( xxf 
 
Resp: 
6
5
1
3
)´(
x
x
xf



 
l) 
14
1
)(


x
xf
 
Resp: 
3)14(
2
)´(



x
xf
 
m) 
14
2
)(


x
xf
 
Resp: 
3)14(
4
)´(



x
xf
 
n) 
3)31(6
1
)(
x
xf


 
Resp: 
4)31(2
1
)´(
x
xf



 
o) 
4)13(4
3
)(


x
xf
 
Resp: 
5)13(
9
)´(



x
xf
 
 
 
 
3) Derive e simplifique. 
a) 
24 )1()( xxxf 
 
Resp: 
)1(2)1(4)´( 423 xxxxxf 
 
b) 
)21()13()( 3 xxxf 
 
Resp: 
32 )13(2)21()13(9)´(  xxxxf
 
c) 
24 )1()13()( xxxf 
 
Resp:
)1()13(2)1()13(12)´( 423 xxxxxf 
 
d) 
12)(  xxxf
 
Resp: 
12
13
12
12)´(





x
x
x
x
xxf
 
e) 
3)1(13)(  xxxf
 
Resp: 
2)1(3
132
3
)´( 

 x
x
xf
 
f) 
3 13)(  xxxf
 
Resp: 
3 2
3
)13(
13)´(


x
x
xxf
 
g) 
12)( 2  xxxf
 
Resp:
12
25
12
122)´(
22





x
xx
x
x
xxxf
 
h) 
124)( 4  xxxxf
 
Resp:
12
1
4
)1(2
)´(
4
3





xxx
x
xf
 
i) 
72 )14()14()(  xxxf
 
Resp: 
6)14(28)14(8)´(  xxxf
 
 
4) Usando as fórmulas de derivação, derive e simplifique quando possível. 
a) 
xexf 5)( 
 
Resp:f´(x)=5e
5x 
b) 
xexf 5)( 
 
Resp:f´(x)=-5e
-5x 
 4 
c) 
xexf )(
 
Resp:f´(x)=-e
-x 
d) 
2
)(
xx ee
xf


 
Resp: 
2
)´(
xx ee
xf


 
e) 
xx
x
ee
e
xf

)(
 
Resp:
2)(
2
)´(
xx ee
xf


 
f) 
xx
xx
ee
ee
xf




)(
 
Resp: 
2)(
4
)´(
xx ee
xf



 
g) 
xx
xx
ee
ee
xf
22
22
)(





 
Resp: 
222 )(
8
)´(
xx ee
xf



 
h) 
)3ln()( xxf 
 
Resp: 
x
xf
1
)´( 
 
i) 
)3ln()( xxf 
 
Resp: 
x
xf
1
)´( 
 
j) 
)3ln()( 3 xxxf 
 
Resp:
22 )3ln(3)´( xxxxf 
 
k) 
)3ln()( 3 xxxf 
 
Resp: 
xx
x
xf
3
)1(3
)´(
3
2



 
l) 
)3ln()( xxf 
 
Resp: 
x
xf
1
)´( 
 
m) 
)3ln()( xxf 
 
Resp: 
x
xf
1
)´( 
 
n) 
)3ln()( 3 xxxf 
 
Resp:
22 )3ln(3)´( xxxxf 
 
o) 
)3ln()( 3 xxxf 
 
Resp: 
xx
x
xf
3
)1(3
)´(
3
2



 
5) Usando as fórmulas de derivação, derive e simplifique quando possível. 
a) 
)45sen()( 2  xxf
 
Resp: f´(x)=10xcos(5x
2
-4) 
b) 
)5sen()( xxf 
 
Resp:f´(x)=5cos(5x) 
c) 
)3sen()( xxf 
 
Resp: f´(x)=-3cos(-3x) 
d) 
)2cos()( xxf 
 
Resp: f´(x)=-2sen(2x) 
e) 
)2cos()( xxf 
 
Resp: f´(x)=2sen(-2x) 
f) 
)2sec(cos)( xxf 
 
Resp: f´(x)=2cossec(-2x)cotg(-2x) 
g) 
)5sen()( 2 xxxf 
 
Resp: 
)5cos(5)5(2)´( 2 xxxxsenxf 
 
h) 
x
x
xf
)5sen(
)( 
 
Resp: 
2
)5()5cos(5
)´(
x
xsenxx
xf


 
i) 
)]5cos()5[sen()( 2 xxexf x 
 
Resp: 
)]5cos(3)5(7[)´( 2 xxsenexf x 