Solução_Calculo_Stewart_6e

F.
TX.10
CHAPTER 3 REVIEW ¤ 131
1. True. This is the Sum Rule.
3. True. This is the Chain Rule.
5. False.
√
d
√ f 0 x
dx f x =
2 √ by the Chain Rule.
x
7. False.
9. True.
d
dx 10x =10 x ln 10
d
dx (tan2 x)=2tanx sec 2 x,and d
dx (sec2 x)=2secx (sec x tan x) =2tanx sec 2 x.
Or:
d
dx (sec2 x)= d
dx (1 + tan2 x)= d
dx (tan2 x).
11. True. g(x) =x 5 ⇒ g 0 (x) =5x 4 ⇒ g 0 (2) = 5(2) 4 =80,andbythedefinition of the derivative,
g(x) − g(2)
lim
= g 0 (2) = 80.
x→2 x − 2
1. y =(x 4 − 3x 2 +5) 3 ⇒
y 0 =3(x 4 − 3x 2 +5) 2
d
dx (x4 − 3x 2 +5)=3(x 4 − 3x 2 +5) 2 (4x 3 − 6x) =6x(x 4 − 3x 2 +5) 2 (2x 2 − 3)
3. y = √ x + 1
3√
x
4 = x1/2 + x −4/3 ⇒ y 0 = 1 2 x−1/2 − 4 3 x−7/3 = 1
2 √ x − 4
3 3√ x 7
5. y =2x √ x 2 +1 ⇒
y 0 =2x · 1
2 (x2 +1) −1/2 (2x)+ √ x 2 +1(2)= √ 2x2
x2 +1 +2√ x 2 +1= 2x2 +2(x 2 +1)
√ = 2(2x2 +1)
√
x2 +1 x2 +1
7. y = e sin 2θ ⇒ y 0 = e sin 2θ d
dθ (sin 2θ) =esin 2θ sin 2θ
(cos 2θ)(2) = 2 cos 2θe
9. y = t
1 − t 2 ⇒ y 0 = (1 − t2 )(1) − t(−2t)
(1 − t 2 ) 2 = 1 − t2 +2t 2
(1 − t 2 ) 2 = t2 +1
(1 − t 2 ) 2
11. y = √ x cos √ x ⇒
y 0 = √
x cos √ x
0
+cos
√
x
√
x
0
=
√
x
− sin √ x
=
− √ 1 2 x−1/2 x sin √ x +cos √
x = cos √ x − √ x sin √ x
2 √ x
1
2 x−1/2
+cos √ x
1
2 x−1/2
13. y = e1/x
⇒ y 0 = x2 (e 1/x ) 0 − e 1/x x 2 0
= x2 (e 1/x )(−1/x 2 ) − e 1/x (2x)
= −e1/x (1 + 2x)
x 2 (x 2 ) 2 x 4 x 4