lista de exercicios 2 de calculo I

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Exerc´ıcios - Matema´tica I - 1o ano Agronomia
Encontre as derivadas das func¸o˜es abaixo usando a regra da cadeia:
1. y = (1 + 3x− 5x2)30
2. y =
(
2x+ 1
3
)3
3. y = (3 + 2x2)4
4. y =
√
1− x2
5. y = 3
√
1 + x3
6. y = (3− 2sen x)5
7. y = 2x+ 5 cos3 x
8. y = tgx−
1
3
tg3 x+
1
5
tg5 x
9. y =
x
1− x
10. y = ln
(
x
1− x
)
11. y = ln
(
1− x
1 + x
)
12. y = ln(sen x)
13. y = sen(x2 − 5x+ 1)
14. y = cos(2x+ 7)
15. y = sen2(x3)
16. y = e−x
2
17. y = 1− e−2x
18. y = ex cos x
19. y = e−x
20. y =
2
1 + e−5x
21. y =
1
√
2pi
e
−
x
2
2
22. y =
e
x + e−x
2
23. y =
e
x − e−x
2
24. y = ln
(
1 + x
1− x
)
25. y = sen(e−x) + cos(ex)
1
Respostas dos exerc´ıcios
1. y′ = 30(1 + 3x− 5x2)29(3− 10x)
2. y′ = 2
(
2x+ 1
3
)2
3. y′ = 16x(3 + 2x2)3
4. y′ = −x(1 − x2)−
1
2 = −x(
√
1− x2)−1 = −
x
√
1− x2
5. y′ = x2( 3
√
1 + x3)−
2
3 =
x
2
3
√
(1 + x3)2
6. y′ = −10 cos x(3− 2sen x)4
7. y′ = 2− 15 cos2 x sen x
8. y′ = sec2 x+ tg2 x sec2 x+ tg4 x sec2 x
9. y′ =
1
(1− x)2
10. y′ =
1(
x
1−x
) 1
(1− x)2
=
1
x(1− x)
11. y′ =
1(
1−x
1+x
) −2
(1 + x)2
= −
2
(1− x)(1 + x)
= −
2
1− x2
12. y′ =
1
sen x
cos x =
cos x
sen x
= cotg x
13. y′ = cos(x2 − 5x+ 1)(2x− 5)
14. y′ = −2sen(2x+ 7)
15. y′ = 6x2sen(x3) cos(x3)
16. y′ = −2xe−x2
17. y′ = 2e−2x
18. y′ = ex cos x− exsen x
19. y′ = −e−x
20. y′ = 10e−5x(1 + e−5x)−2 =
10e−5x
(1 + e−5x)2
21. y′ = −
xe
−
x
2
2
√
2pi
22. y′ =
e
x − e−x
2
23. y′ =
e
x + e−x
2
24. y′ =
1(
1+x
1−x
) 2
(1− x)2
=
2
(1 + x)(1 − x)
=
2
(1− x2)
25. y = −e−x cos(e−x)− exsen(ex)
2