Thomas Calculus 13th [Solutions]

Section 3.6 The Chain Rule 159
11 2 dy 11 3 10
dy
3
(f ) y ( x f ( x))
2( x f ( x)) 11 x f ( x)
2(1 f (1)) (11
dx
dx x 1
f (1))
2(1 3)
3
11 1 2 32 1
3 3
4 3 3
(g)
dy
dy
y f ( x g( x))
f ( x g( x))(1 g ( x ))
f (0 g(0))(1 g (0)) f (1) 1 1
dx
dx x 0
3
1 4 4
3 3 9
89. ds ds d :
dt d dt
s cos ds sin ds
d
d 3
2
sin 3 1 so that ds ds d
2
dt d dt
1 5 5
dy dy
90. dx
:
dt dx dt
y
2
x
7x
5
dy
dx
dy
dx x 1
2 x 7 9 so that dy dy dx
dt dx dt
9
1
3
3
dy
91. With y x , we should get 1 for both (a) and (b):
dx
(a) y u dy
7
1
;
5 du 5
u 5 x 35 du
dx
5; therefore, dy dy du 1
5 1, as expected
dx du dx 5
(b)
1 dy
1
1
1
y
; u ( 1)
du
2
x ( x 1) (1)
1
; therefore dy dy du
2
2
u du u
1 1
1
2 2
( x 1) ( x 1)
2 1
( x 1)
2
dx
( x 1) 1, again as expected
( x 1)
1 1
dx du dx 2
u ( x
2
1)
3/2
92. With y x , we should get
3 1/2
x for both (a) and (b):
dx 2
3 dy 2
(a) y u 3 u ; u x du 1
; therefore,
du
as expected.
dy
(b) y u
1
du
2
again as expected.
u
dy
dx
; u x 3 x ; therefore,
2
x
3 du 2
dx
dy dy du
dx du dx
3u
2
dy dy du 1
dx du dx 2 u
1 2
x
1 3
2 x
2 x 2
3x
2
3( ) x ,
2
1
x
3
2 3 1/2
x
2
3 x ,
93.
y
2
x 1
and
x 1
4(0 1)
x 0 3
(0 1)
y
3
2
0 1
0 1
2
x 1 ( x 1) 1 ( x 1) 1
x 1 ( x 1)
x 0 y ( 1) 1. y 2
2
4
1
4 y 1 4( x 0) y 4x
1
( 1) 4( 1)
2 x 2 x
( x 1) ( 1)
2 ( 1)
3
x x
94.
2
y x x 7 and x 2
2(2) 1
y
3 1
x 2
2
2 (2) (2) 7
6 2
y
2
(2) (2) 7 9 3. y
y 3
1
( x 2) y
1
x 2
2 2
1
2
2
1/2
x x 7 (2x
1)
2x
1
2
2 x x 7
95.
y
(a)
(b)
2 tan
dy
x
4
dy
dx
2
2 4
2sec
2 x
2
sec
x
4 4 2 4
sec ( ) slope of tangent is ; thus, y (1) 2 tan 2 and y (1) tangent line is
dx x 1
4
given by y 2 ( x 1) y x 2
2
y sec
x
and the smallest value the secant function can have in 2 x 2 is 1 the minimum
2 4
value of y is 2
and that occurs when
2 x
2 x
2 2 4 4
sec 1 sec 1
x
4
sec x 0.
96. (a) y sin 2x y 2cos 2 x y (0) 2cos(0) 2 tangent to y sin 2x at the origin is y 2 x;
y sin
x
y
1
cos
x
y (0)
1
cos 0
1
tangent to y sin
x at the origin is
1
2 .
2 2 2 2 2
y x The tangents are perpendicular to each other at the origin since the product of their slopes is 1.
2
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