Thomas Calculus 13th [Solutions]

Section 3.5 Derivatives of Trigonometric Functions 145
dy
10. y (sin x cos x)sec
x (sin x cos x ) d (sec x) sec x d (sin x cos x)
dx
dx
dx
(sin x cos x)sin x
(sin x cos x)(sec x tan x) (sec x)(cos x sin x)
cos x sin x
2
cos x cos x
2 2
sin x cos xsin x cos x cos xsin
x 1 2
sec x
2
2
cos x
cos x
2
Note also that y sin xsec x cos xsec x tan x 1 dy sec x.
dx
d
d
11.
cot x dy (1 cot
y ) (cot ) (cot ) (1 cot )
dx
dx
1 cot x dx
2
(1 cot x)
2
csc x
2
csc x cot x
2
csc x cot x
2
csc x
2
(1 cot x)
2
(1 cot x)
2 2
(1 cot x)( csc x) (cot x)( csc x)
2
(1 cot )
x
12. y cos x dy (1 sin x d
d
) (cos ) (cos ) (1 sin )
dx
x x dx
x
1 sin x dx
(1 sin x)
2
sin x 1 (1 sin x) 1
2
2
(1 sin x)
(1 sin x)
1 sin x
x x x x
(1 sin )( sin ) (cos )(cos )
2
(1 sin )
x
2 2
sin sin cos
2
(1 sin )
x x x
x
13. y 4 1
cos x tan x
4sec x cot x
dy
dx
2
4sec x tan x csc x
14. y cos x x dy x( sin x) (cos x)(1)
x cos x dx
x
2
x x x
(cos )(1) ( sin )
2
cos
x
x sin x cos x cos x xsin
x
2
x
2
cos x
15. y (sec x tan x ) (sec x dy
tan x )
dx
(sec x tan x) d
dx
(sec x tan x ) (sec x tan x ) d
dx
(sec x tan x )
2
2
(sec x tan x)(sec x tan x sec x ) (sec x tan x)
(sec x tan x sec x)
2 2 3 2 2 2 3 2
(sec x tan x sec x tan x sec x sec x tan x) (sec x tan x sec x tan x sec x tan x sec x) 0.
2 2 2 2 dy
Note also that y sec x tan x (tan x 1) tan x 1 0.
dx
16.
2 dy 2
y x cos x 2xsin x 2cos x ( x ( sin x) (cos x)(2 x )) (2x cos x (sin x)(2)) 2( sin x)
dx
x
2 sin x 2x cos x 2x cos x
2
2sin x 2sin x x sin x
17.
3
f ( x) x sin x cos x f ( x) x sin x( sin x)
3 2 3 2 2
x sin x x cos x 3x sin x cos x
3
3 2
x cos x(cos x) 3x sin x cos x
2
18. g( x) (2 x)
tan x g ( x) (2 x)
(2 tan x sec x)
2
2(2 x) tan x(sec x tan x)
2
2
2 2
( 1) tan x 2(2 x)
tan xsec x tan x
19. s tan t e
t
ds
dt
2
sec
t
t e 20.
2
s t sect 5e t ds 2t sect tant 5e
t
dt
21. s 1 csct
ds (1 csc t)( csct cot t) (1 csc t)(csct cot t)
1 csct
dt
(1 csc t)
2
2 2
csc cot csc cot csc cot csc cot
2
(1 csc )
t t t t t t t t
t
t
t
2csc cot
2
(1 csc )
t
22. s sin t ds (1 cos t)(cos t) (sin t)(sin t)
1 cost
dt
(1 cos t)
2
2 2
cost cos t sin t cost
1
2 2
(1 cos ) (1 cos )
t
t
1 1
1 cos t cost
1
23.
r
2
4 sin dr
d
2
d
d
(sin ) (sin )(2 )
2
( cos 2 sin ) ( cos 2sin )
24. r sin cos dr ( cos (sin )(1))
d
sin
cos
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