Thomas Calculus 13th [Solutions]

Section 3.3 Differentiation Rules 135
48.
u
2 2
( x x)( x x 1)
du
dx
4
x
4
0 3x
2
3 4
x( x 1)( x x 1) x( x 1) x x
4
4 4
x
x x
4
2
5
3x
3 d u 12x
12
4 2 5
x dx x
1 x 1
4
x
x
3
49. w 1 3z
(3 z)
1 1
z 1 (3 z)
3z
3
2
d w 3 3
2z
0 2z
2
2 3
dz
z
z
1 1
3
3
z
z
1 8
3
z
dw
dz
2 2
z 0 1 z 1 1 1
2
z
50.
p
q
2
3
3 3
( q 1) ( q 1)
2
d p 3
q 1
2 3
dq
q
q
3 2 3 2
2
( q 3q 3q 1) ( q 3q 3q
1)
3
q
2 2
3 q 3
3 2
2q 6q 2 q( q 3)
1 1 q
2q
2
1
dp
dq
1
2
2
1
2q
q
2
51.
52.
2 2z 2 2z 2z 2z 2z 2z 2z
w 3z e dw 3 z (2 e ) 6ze dw 6 ze (1 z) d w 6 ze (1) (1 z)(6 z(2 e ) 6 e )
dz dz 2
dz
2
d w
2
dz
2z
2
6 e (1 4z 2 z ), NOTE:
d 2z
2z
( e ) 2e from part b of Example 6
dz
z 2 z 3 2 z 3 2 2 z z 3 2
w e ( z 1)( z 1) e ( z z z 1) w e ( z z z 1) (3z 2z 1) e e ( z 2 z z)
z 3 2 2 z z 3 2
and w e ( z 2 z z) (3z 4z 1) e e ( z 5z 3z
1)
2
53. u(0) 5, u (0) 3, v(0) 1, v (0) 2
(a) d ( )
dx uv uv vu d ( uv) u(0) v (0) v(0) u (0) 5 2 ( 1)( 3) 13
dx x 0
(b) d u vu uv d u v(0) u (0) u(0) v (0) ( 1)( 3) (5)(2)
7
dx v 2
v dx v 2
2
x 0 ( v(0))
( 1)
(c) d v uv vu d v u(0) v (0) v(0) u (0) (5)(2) ( 1)( 3) 7
dx u 2
u dx u 2
2
x 0 ( u(0))
(5) 25
(d) d (7v 2 u) 7v 2u d (7v 2 u ) |
dx
dx
x 0 7 v (0) 2 u (0) 7 2 2( 3) 20
54. u(1) 2, u (1) 0, v(1) 5, v (1) 1
(a) d ( uv ) |
dx x 1 u (1) (1) v (1) (1) 2 ( 1) 5 0 2
(b) d u v(1) u (1) u(1) v (1) 5 0 2 ( 1) 2
dx v 2 2
x 1 ( v(1)) (5) 25
(c) d v u(1) v (1) v(1) u (1) 2 ( 1) 5 0 1
dx u 2 2
x 1 ( u(1)) (2) 2
(d) d (7v dx
2 u ) | x 1 7 v (1) 2 u (1) 7 ( 1) 2 0 7
55.
3
3
y x 4x 1. Note that (2,1) is on the curve: 1 2 4(2) 1
2
2
(a) Slope of the tangent at ( x, y ) is y 3x 4 slope of the tangent at (2,1) is y (2) 3(2) 4 8. Thus
the slope of the line perpendicular to the tangent at (2,1) is 1 the equation of the line perpendicular to
8
the tangent line at (2, 1) is y 1 1 ( x 2) or y x 5
8
8 4 .
2
(b) The slope of the curve at x is m 3x 4 and the smallest value for m is 4 when x 0 and y 1.
2
2
2
(c) We want the slope of the curve to be 8 y 8 3x
4 8 3x
12 x 4 x 2. When
x 2, y 1 and the tangent line has equation y 1 8( x 2) or y 8x 15; When x 2,
3
y ( 2) 4( 2) 1 1, and the tangent line has equation y 1 8( x 2) or y 8x
17.
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