Calculo1 - Lista07

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Universidade Federal de Sa˜o Carlos-Departamento de Matema´tica
89109-Ca´lculo 1-Turma E: Lista 7
Prof(a) Alessandra Verri 7 de maio de 2017
Exerc´ıcio 1. Derive a func¸a˜o:
(a)
x
x2 + 1
(b)
x2 − 1
x + 1
(c)
3x2 + 3
5x− 3 (d)
√
x
x + 1
(e) 5x +
x
x− 1 (f)
√
x +
3
x3 + 2
(g)
3
√
x + x√
x
(h)
x + 4
√
x
x2 + 3
(i) 3x2 + 5 cosx (j)
cosx
x2 + 1
(k) x senx (l) x2 tgx
(m)
x + 1
tgx
(n)
3
senx + cosx
(o)
secx
3x + 2
(p) cosx + (x2 + 1) sinx (q)
√
x secx
(r) 4 secx + cotgx (s)
x + senx
x− cosx
Exerc´ıcio 2. Derive a func¸a˜o:
(a) x2ex (b) 3x + 5 lnx (c) ex cosx (d)
1 + ex
1− ex (e) x
2 lnx + 2ex (f)
x + 1
x lnx
(g) 4 + 5x2 lnx (h)
ex
x2 + 1
(i)
lnx
x
(j)
ex
x + 1
Exerc´ıcio 3. Sejam f , g e h func¸o˜es deriva´veis. Verifique que:
(f(x)g(x)h(x))′ = f ′(x)g(x)h(x) + f(x)g′(x)h(x) + f(x)g(x)h′(x).
Exerc´ıcio 4. Derive a func¸a˜o:
(a) x ex cosx (b) x2 cosx (1 + lnx) (c) ex senx cosx (d) (1 +
√
x) ex tgx
Respostas: 1. (a)
1− x2
(x2 + 1)2
(b)
x2 + 2x + 1
(x + 1)2
(c)
15x2 − 18x− 15
(5x− 3)2 (d)
1− x
2
√
x(x + 1)2
(e) 5− 1
(x− 1)2 (f)
1
2
√
x
− 9x
2
(x3 + 2)2
(g)
3x− 3√x
6x
√
x
(h)
4
4
√
x3(3− x2)− 7x2 + 3
4
4
√
x3(x2 + 3)2
(i) 6x− 5 senx (j) −(x
2 + 1) senx + 2x cosx
(x2 + 1)2
(k) senx + x cosx (l) x(2 tgx + x sec2x)
(m)
tgx− (x + 1)sec2x
tg2x
(n) −3(cosx− senx)
(senx + cosx)2
(o)
secx(3x tgx + 2tgx− 3)
(3x + 2)2
(p) (2x− 1)senx + (x2 + 1) cosx (q) secx (1 + 2x tgx)
2
√
x
(r) −3 senx + 5 secx tgx
(s)
(x− 1) cosx− (x + 1) senx− 1
(x− cosx)2
2. (a) x ex (2 + x) (b) 3 +
5
x
(c) ex(cosx− senx) (d) 2e
x
(1− ex)2 (e) 2x lnx + x + 2e
x
(f)
−x− lnx− 1
(x lnx)2
(g) 5x (1 + 2 lnx) (h)
ex(x− 1)2
(x2 + 1)2
(i)
1− lnx
x2
(j)
x ex
(x + 1)2
3. (a) ex (cosx + x cosx− x senx) (b) x[(1 + lnx)(2 cosx− x senx) + cosx]
(c) ex [senx cosx + cos2 x− sen2 x] (d) ex
[
tgx
2
√
x
+ (1 +
√
x)(tgx + sec2x)
]
2