Thomas Calculus 13th [Solutions]

Section 3.6 The Chain Rule 153
2 2
15. With u tan x , y sec u dy dy
: du (sec tan )(sec ) (sec(tan ) tan(tan ))sec
dx du dx
u u x x x x
16. With 1 dy dy
u , y cot u: du ( csc 2 u) 1 1 csc
2 1
x dx du dx 2 2
x x
x
17. With
u 3 2 2 2 2
tan x , y u dy dy
: du 3 sec 3tan sec
dx du dx
u x x x
4 5 5
18. With u cos x , y 5 u dy dy
: du ( 20 )( sin ) 20(cos )(sin )
dx du dx
u x x x
19. With u = 5x,
y e u dy dy
5
: du u x
( 5) 5
dx du dx
e e
20. With
u 2 x
2 2 2 /3
, u dy dy
: u x
3 y e du
dx du dx
e 3 3
e
21. With u = 5 7x,
y e u dy dy
5 7
: du u x
( 7) 7
dx du dx
e e
22. With
2 u dy dy u 1 1/2
2
4
u 4 x x , y e : du e 4 x 2x 2x e
dx du dx 2
x
x
x
2
23.
24.
1/2 dp 1 1/2 1 1/2
p 3 t (3 t) (3 t) d (3 t) (3 t)
1
dt 2 dt 2 2 3 t
q 3 2 2 1/3 1 2 2/3 2
2 r r (2 r r dq
) (2 ) d (2 )
dr 3
r r dr
r r 1 2 2/3
2 2
3 (2 r r ) (2 2 r) r
3(2 r r
2 ) 2/3
25. s 4 sin 3t 4 cos5t ds 4 cos3 t d (3 t) 4 ( sin 5 t) d (5 t) 4 cos3t 4 sin 5t 3 5 dt 3 dt 5
dt
4 (cos3t
sin 5 t)
26. s sin 3 3 3 3 3 3
2 2 2 2 3 cos 3 t 3 sin 3 t
dt dt 2 dt 2 2 2 2 2
3 cos 3 t sin 3 t
2 2 2
27.
r 1 2
2
(csc cot ) dr (csc cot ) d (csc cot ) csc cot csc
2
2
d
d
(csc cot )
csc (cot csc ) csc
(csc cot ) csc cot
28.
r
3/2 3
1/2
6(sec tan ) dr 6 (sec tan ) d (sec tan )
d 2
d
2
9 sec tan (sec tan sec )
29.
30.
2 4 2 dy 2 4 4 2 2 2
y x sin x x cos x x d (sin x) sin x d ( x ) x d (cos x) cos x d ( x)
dx dx dx dx dx
x 2 3 4 3 2
(4sin x d (sin )) 2 sin ( 2cos d (cos )) cos
dx
x x x x x dx
x x
2 3 4 3 2
x (4sin x cos x) 2xsin x x(( 2cos x) ( sin x)) cos x
2 3 4 3 2
4x sin x cos x 2xsin x 2x sin x cos x cos x
1 5 3 1 5 5 1
3 3
y sin x x cos x y d (sin x) sin x d x d (cos x) cos x d ( x)
x 3 x dx dx x 3 dx dx 3
1 6 5 2 3
( 5sin x cos x) (sin x) 1 x ((3cos x)( sin x)) (cos x)
1
x
2
x 3 3
5 6 5 2 3
sin x cos x 1 sin x x cos xsin x 1 cos x
x
2
x
3
Copyright
2014 Pearson Education, Inc.