HOMEWORK #9 – SOLUTIONS Page 214 46) 48) 52) y = tan −1 x
1
y ' = tan −1 x
2
(
)
−1
2
⋅
1
1
=
2
1+ x
2 tan −1 x 1 + x 2
(
)
g ( x ) = x 2 − 1 sec −1 x
g '( x) = x2 − 1 ⋅
1
x x2 − 1
+
(
)
1 2
x −1
2
−1
2
⋅ 2x ⋅ sec −1 x = 1 x sec −1 x
+
x
x2 − 1
F (θ ) = arcsin sin θ
F ' (θ ) =
−1
1
1
cos θ
⋅ ( sin θ ) 2 cos θ =
1 − sin θ 2
2 sin θ (1 − sin θ )
Page 220 4) f ( x ) = ln sin 2 x
1
2 cos x
f '( x) =
⋅
2
sin
x
cos
x
=
=
2
cot
x
sin 2 x
sin x
6) f ( x ) = log 5 xe x
1
x +1 f '( x) = x
⋅ xe x + e x =
xe ln 5
x ln 5
(
)
( )
(
)
20) 30) 38) (
)
y = ⎡⎣ ln 1 + e x ⎤⎦
2
(
2e x ln 1 + e x
1
x
y ' = 2 ln 1 + e ⋅
⋅e =
1 + ex
1 + ex
(
x
)
f ( x ) = ln ln ln x
1
1 1
⋅
⋅
ln ln x ln x x
D = { x : ln ln x > 0} = { x : ln x > 1} = { x : x > e}
f '( x) =
(
2
)
y = xe x x 2 + 1
10
1
2
2
ln y = ln x + ln e + ln x + 1 = ln x + x ln e + 10 ln x + 1
2
1
1 1
1
1
20x
y ' = ⋅ + 2x + 10 ⋅ 2
⋅ 2x =
+ 2x + 2
=
y
2 x
x +1
2x
x +1
(
x2
(
)
)
x 2 + 1 + 2x ⋅ 2x ⋅ x 2 + 1 + 2x ⋅ 20x
(
)
(
10
2
2x x 2 + 1
41x 2 + 4x 4 + 4x 2 + 1
=
= 2x x 2 + 1
(
4x 4 + 45x 2 + 1
2x x 2 + 1
(
)
(
2
)
y ' = xe x x 2 + 1
e
x2
(x
2
) ( 4x
+1
9
4
2 x
40) ) 10
4x 4 + 45x 2 + 1
=
2x x 2 + 1
(
)
+ 45x + 1
2
)
)
)
x2 + 1
x2 − 1
1
ln y = ⎡⎣ ln x 2 + 1 − ln x 2 − 1 ⎤⎦
4
1
1⎡ 1
1
⎤
y' = ⎢ 2
⋅ 2x − 2
⋅ 2x ⎥ =
y
4 ⎣x +1
x −1
⎦
y=
4
(
(
x
)
2 x +1
48) y' =
2
4
−
)
(
x
(
)
2 x −1
2
=
)
(
)
y = ( ln x )
(
(
)(
) (
cos x
1
1 1
y ' = cos x ⋅
⋅ − sin x ⋅ ln ln x y
ln x x
cos x
)
)(
)
4 x +1 x −1
2
2
x2 + 1
−2x
−x
⋅
= 4
2
2
2
x −1 2 x +1 x −1
x −1
y ' = ( ln x )
(
2x x 2 − 1 − 2x x 2 + 1
ln y = cos x ln ln x
⎛ cos x
⎞
− sin x ln ln x ⎟
⎜⎝
⎠
x ln x
)
4
x2 + 1
x2 − 1