derivadas

Quest˜ ao 1. Encontre a derivada da fun¸ca˜o dada usando a defini¸ca˜o. Diga quais s˜ao os dom´ınios da fun¸c˜ao e da derivada.
√
(d) f (x) = mx + b. f (x) = m; R; R
(a) f (x) = x + x. f (x) = 1 + 2√1 x ; [0, ∞); (0, ∞)
(b) f (x) = 1, 5x2 − x + 3, 7. f (x) = 3x − 1; R; R
(c) f (x) =

3+x
.
1 − 3x

f (x) =

10
;R
(1−3x)2

− { 13 }; R − { 13 }

1
1
(e) f (x) = x − .
2
3
1
(f) g(t) = √ . t f (x) = 12 ; R; R

g (t) =

Quest˜ ao 2. Se g(x) = x2/3 , mostre que g (0) n˜ao existe. Se a = 0, encontre g (a).

−1
√ ; (0, ∞); (0, ∞)
2t t

2
3a1/3

Quest˜ ao 3. Utilizando a regra do produto, calcule a derivada das fun¸c˜oes abaixo:
(a) y = (2x − 3)(x2 − 5x) y = 2(x2 − 5x) + (2x − 3)(2x − 5) = 6x2 − 26x + 15
(b) w = (t − 1)(t + 3) w = 2t + 2
(c) p = (t2 − 5)(2t + 3) p = 6t2 + 6t − 10

Quest˜ ao 4. Derive:
(a) y = 3x6 + 9x − 3
5

(b) y = x− 9

y =−

√
9
7
(c) y = 10 x6 − √ x √
5
7
(d) y = x x2 + 4 √ x x

y = 18x5 + 9
5
9x

14
9

y =

60
√
77x

y =

9

√
7

+

x2
7

√9
2 x3

−

√45
2 x11

Quest˜ ao 5. Utilizando a regra do quociente, determine a derivada das fun¸co˜es abaixo:
2x − 3
13
(x+5)2 x+5 t2 − 2t
3t2 +8t−8
(b) w =
(3t+4)2
3t + 4
5
−10t+15
(c) p = 2
(t2 −3t+5)2 t − 3t + 5

(a) y =

Quest˜ ao 6. Utilizando a regra da cadeia, calcule a derivada das fun¸co˜es abaixo:
(a) y = sin(4x)
(b) y = e3x
(c) y = esin t
(d) y =

√

3.e3x

3x + 1

√

x + ex

(j) y = cos(5x) −5. sin(5x)
(l) y = sin(t3 ) 3t2 . cos(t3 )

cos t.esin t

(e) y = ln(2t + 1)
(f) y =

4. cos(4x)

(m) y = cos(ex ) −ex . sin(ex )

√3
2 3x+1

(n) y = etg x

2
2t+1

etg x . sec2 x

(o) y = (t2 + 3)4

1+ex
√
2 x+ex

8t.(t2 + 3)3

(p) y = tg(3x) 3. sec2 (3x)

(g) y = cos(1 − x2 ) 2x. sin(1 − x2 )

(q) y = sin(cos x) − sin x. cos(cos x)

(h) y = (sin x + cos x)3

(r) y = sec(3x) 3 sec(3x). tg(3x)

(i) y = ln(t2 + 3t + 9)

3(sin x + cos x)2