Thomas Calculus 13th [Solutions]

Section 3.6 The Chain Rule 161
sin 2( x h) sin 2x
105. As h 0, the graph of y
h
approaches the graph of y 2cos 2x because
sin 2( x h) sin 2x
lim
d
(sin 2 x) 2cos 2 x.
h
0
h
dx
2 2
cos[( x h) ] cos( x )
106. As h 0, the graph of y
h
2
approaches the graph of y 2x sin ( x ) because
2 2
cos[( x h) ] cos( x )
2 2
lim
d
[cos ( x )] 2xsin ( x ).
h
0
h
dx
107. From the power rule, with
y x 1/4 , we get
dy 1 d
1 1 1
dx
2 x
dx
2 x 2 x 4
dy
dx
3/4
1
4
3/4
x x , in agreement.
x . From the chain rule, y x
108. From the power rule, with
109. (a)
y x 3/4 , we get
dy
dx
3
4
1/4
x . From the chain rule, y x x
dy 1 d
x x
dx
2 x x
dx
dy 1 1 1 3 3 x 3 x 3
dx
2 x x 2 x 2 x x
2
4 x x 4 x x
4
x x x x
1/4
, in agreement.
df
dt
(b) 1.27324sin 2t 0.42444sin 6t 0.2546sin10t
0.18186sin14t
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