RESPOSTAS DOS EXERCÍCIOS DE REGRA DA CADEIA

Prévia do material em texto

SEÇÃO 3.4 REGRA DA CADEIA  1
 1. 10(x2 + 4x + 6)4 (x + 2) 2. 3 sec2 3x
 3. –sen(tg x) sec2 x 4. 
2
3 2/3(1 )+
x
x
 5. F¢(x) = 4(x3 – 5x)3 (3x2 – 5)
 6. f ¢(t) = –16(2t2 – 6t + 1)–9 (2t – 3)
 7. 
2
2 7
( )
2 7
x
g x
x x
-¢ =
-
 8. 
2 5
8(1 )
( )
( 2 5)
-¢ = - -
t
f t
t t
 9. 1/2 23
2
( ) ( 1/ ) (1 1/ )¢ = - +h t t t t
 10. 2
1 1
 cosy
xx
¢ = -
 11. G¢(x) = 6(3x – 2)9 (5x2 – x + 1)11 (85x2 – 51x + 9)
 12. g¢(t) = 12t(6t2 + 5)2 (t3 – 7)3 (9t3 + 5t – 21)
 13. 
2
4
39( 6)
( )
( 7)
y
F y
y
-¢ = +
 14. 
3/43 2
3 3 2
1 1 3
( )
2 1 ( 1)
-æ ö+ -÷ç ÷¢ ç= ÷ç ÷÷ç - -è ø
t t
s t
t t
 15. 6/52
5
( ) (2 1)-¢ = - -f z z 16. 3/2
14 3
( )
2(7 3 )
-¢ = -
x
f x
x
 17. y¢ = 5–1/x (ln 5)/x2 18. 
2sec
1 2 tg 
x
y
x
¢ = +
 19. y¢ = 3 sen x cos x (sen x – cos x)
 20. y¢ = – k sen kx sen (2 cos kx)
 21. 
3 4
2
3 2
(1 )
+¢ = +
x x
x
e e
y
e
 22. y¢ = 5 cos (5q) esen 5q
 23. ( ) 22 12 2cos 12 sen 1 1
- +¢ = +
+
x
y x x
x
 24. y¢ = 0
 25. f ¢(x) = 9[x3 + (2x – 1)3]2 (9x2 – 8x + 2)
 26. g¢(t) = [(1 – 3t)4 + t 4]–3/4 [t3 – 3 (1 – 3t3)]
 27. ( )2
2 1 1
 sen cos
1 11
x x
y
x xx x
æ ö æ ö- -÷ ÷ç ç÷ ÷¢ ç ç= ÷ ÷ç ç÷ ÷÷ ÷ç ç+ +è ø è ø+
 28. 
2 2
2
1
( 1) sec
1
2 1 tg
x x
x
y
x x
x
æ ö÷ç- + ÷ç ÷çè ø¢ = æ ö÷ç+ + ÷ç ÷çè ø
 29. p¢(t) = –2[(1 + 2/t)–1 + 3t]–3 [2(t + 2)–2 + 3]
 30. ( )
( )
7
3
2/3
1
3
( ) 8 2 9
1
1 2 9 1
2 9
N y y y y
y y
y
-
¢ = + + -
é ùæ ö÷çê ú÷ç+ + - + ÷ê úç ÷ç ÷-è øê úë û
 31. F¢(s) = 24(3s + 5)7, F ¢¢(s) = 504(3s + 5)6
 32. 3/2 5/2312 4( ) (1 ) , ( ) (1 )g u u g u u
- -¢ ¢¢= - = -
 33. 2 1/4 2 5/4 23 3
2 4
(1 ) , (1 ) ( 2)y x x y x x- -¢ ¢¢=- - = - -
 34. f ¢(x) = –10 cossec2(5x) cotg(5x),
 f ¢¢(x) = 50 cossec2(5x) [2 cotg2(5x) + cossec2(5x)]
 35. 
2 3 1/2
sec tg
( ) ,
2
sec tg sec sec tg
( )
4
r r
F r
r
r r r r r r
F r
r
-
¢ =
+ -¢¢ =
 36. H¢(t) = 6 tg2(2t – 1) sec2(2t – 1),
 H¢¢(t) = 24 tg(2t – 1) sec4(2t – 1)
 + 24 tg3(2t – 1) sec2(2t – 1)
 37. 3 11
16 4
y x= - +
 38. 3 3
2 12
1y x p= - + +
 39. y = 10[(x – 1) ln 10 + 1] 40. y = 0
 41. 2=y 42. y = 39x – 80 43. 4x + y = p + 1
 44. 
{ }
2
6
( ) 2 sec 3 (1 3 tg 3 ),
(2 2) , um inteiro (tanto quanto )
f x x x x x
x x n n f fp
¢ = +
¢¹ -
 45. 
) ( )1 12 2
cos 2 1
( ) ,
2 1
dom( ) , , dom( ) , 
x
f x
x
f f
+¢ = +
é ¢= - ¥ = - ¥êë
 46. 
{
{ }}
{
{ }}
2
2 2
2
2 2
sen
( ) ,
4 cos
dom( ) 0 /4 ou
[(4 1) /2] [(4 1) /2]
para algum 1, 2, 3, ,
dom( ) 0 /4 ou
[(4 1) /2] [(4 1) /2]
para algum 1, 2, 3, 
x
f x
x x
f x x
n x n
n
f x x
n x n
n
p
p p
p
p p
¢ = -
= £ £
- £ £ +
Î
¢ = < <
- < < +
Î


 47. 
{
}
{
}
sen sen
( ) ,
2 2 cos
dom( ) 0 /2 ou
(4 1) /2 (4 1) /2
para algum 1, 2, 3, ,
dom( ) 0 /2 ou
(4 1) /2 (4 1) /2
para algum 1, 2, 3, 
x x
f x
x x
f x x
n x n
n
f x x
n x n
n
p
p p
p
p p
¢ = - -
= £ £
- £ £ +
=
¢ = < <
- < < +
=


3.4 RESPOSTAS