Thomas Calculus 13th [Solutions]

156 Chapter 3 Derivatives
58.
sin( t/2) 3 dy sin( t/2) 2 sin( t/2) 1 3 sin( t /2) 2sin( t/2) 3 3sin( t/2)
y ( e ) 3( e ) e cos t cos t e e cos t e
dt
2 2 2 2 2 2
59.
y
2
t
3
t 4t
3
dy
dt
3
t 2 ( t 4 t )(2 t ) t (3 t 4) 3 t 2 t 8 t 3 t 4 t
3 3 2 3 2 3 2
t 4 t ( t 4 t) ( t 4 t) ( t 4 t)
2 3 2 2
4 4 2 4 2
4 4 2
3 t ( t 4 t )
4 2 4
t ( t 4 t)
2 2
3 t ( t 4)
2 4
( t 4)
60.
3t
4
5t
2
5
dy
dt
6
3t
4 (5 t 2) 3 (3 t 4) 5
5t
2 (5t
2)
y 5
2
6
5 5 t 2 15 t 6 15 t 20
3t
4 2
(5t
2)
6
(5 2)
5 t 26
(3t 4) (5t
2)
6 2
130(5t
2)
6
(3t
4)
4
61. y sin (cos (2t 5))
dy cos(cos (2t 5)) d cos (2t 5) cos(cos (2t 5)) ( sin (2t 5)) d (2t
5)
dt dt dt
2cos(cos(2t
5))(sin(2t
5))
62.
y
cos 5sin t dy
sin 5sin t d 5sin t sin 5sin t 5cos t
3 dt 3 dt 3
3 3
5
3 sin 5sin t cos t
3 3
d
dt
t
3
63.
4
3
1 tan t dy
4
2
y
3 1 tan t d 4
1 tan t
4
2
3 1 tan t 3
4 tan t d tan t
12 dt
12 dt
12
12
12 dt 12
4
2
12 1 tan t 3 2
tan t sec t 1
4
2
3 2
1 tan t tan t sec t
12
12 12 12
12 12 12
64.
y
2
3
1
6 1 cos (7 ) dy
t
dt
3 2 2
[1 cos (7 )] 2cos(7 )( sin(7 ))(7)
6 t t t 2
2
7 1 cos (7 t) (cos(7 t)sin(7 t))
65.
y
2 1/2
(1 cos ( t ))
1
2
2 1/2
(1 cos ( ))
dy 1
2 1/2
dt 2 (1 cos ( t )) d
dt
2
t sin ( t )
(sin ( t )) 2t
2
1 cos ( t )
t
2
2 1
2 1/2
(1 cos ( t )) (1 cos ( t ))
2
2 2
sin( t ) d ( t )
dt
66. y dy
4sin 1 t 4cos
dt
1 t d
dt
1 t 4cos 1 t 1 d 1
2 1 t
dt
2cos 1 t cos 1 t
t
1 t 2
t
t t
t
67.
y
2 3
tan (sin t)
dy
dt
3 2 3 2
2 tan(sin t) sec (sin t) (3sin t (cos t))
3 2 3 2
6 tan(sin t)sec (sin t)sin t cost
68.
4 2 dy 3 2 2
y cos (sec 3 t) 4cos sec (3 t) sin(sec (3 t) 2 sec(3 t) sec(3 t) tan(3 t) 3
dt
3 2 2 2
24cos sec (3 t) sin sec (3 t) sec (3 t) tan(3 t)
69.
2 4
y 3 t (2t
5)
dy
dt
2 3 2 4
3t 4(2t 5) (4 t) 3 (2t
5)
2 3 2 2
3(2t 5) [16t 2t
5]
2 3 2
3(2t
5) (18t
5)
70. y 3t 2 1 t
dy
dt
1 3 1 1
2 3t
2 1 t 2 2 1 t 2 1 t
t t 1/2
3 1 2 1 t
1/2
2
1
2 3 2 1
1
2 3t
2 1 t
12 1 t 2 1 t 1
4 1 t 2 1 t
1
2
1/2
(1 t ) ( 1)
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