Thomas Calculus 13th [Solutions]

138 Chapter 3 Derivatives
74. (a) We use the Quotient rule to derive the Reciprocal Rule (with u 1): d 1
v 0 1 1
dx
dx v 2
v v
(b) Now, using the Reciprocal Rule and the Product Rule, well derive the Quotient Rule:
d u d
u 1 u
d 1 1 du
(Product Rule) u 1 dv 1 du
(Reciprocal Rule)
2
dx v dx v dx v v dx
dv du du dv
d u
u v v u
dx dx dx dx
,
dx v 2 2
v v
the Quotient Rule.
75. (a)
d
( uvw) d
(( uv) w ) ( uv )
dw
w d
( uv ) uv dw w u dv v du uv
dw
wu
dv
wv
du
dx
dx
uvw uv w u vw
dx
dx
v
dx
dx dx dx
d d du d
dx dx dx dx
d
du4 du3
du2 du1
dx 1 2 3 4 1 2 3 dx 4 1 2 dx 3 1 dx 3 2 dx
4
(b) ( u1u 2u3u4 ) u1u 2u3 u4 u1u 2u3 u4 u1u 2u3
v dx
dv
dx dx dx
u u u u u u u u u u u u u u (using (a) above)
d
du4 du3
du2 du1
u1u 2u3u4 u1u 2u3 u
dx dx 1u2u4 u
dx 1u3u4 u
dx 2u3u4
dx
dv
dx
2 2
v
u1u 2u3u4 u1u 2u3u4 u1u 2u3u4 u1u 2u3u4
(c) Generalizing (a) and (b) above,
d
( 1 )
dx u u n u 1 u 2 u n 1 u n u 1 u 2 u n 2 u n 1 u n u 1 u 2 u n
1 dv
.
dx
m
dx dx x
76.
d
( x )
d 1
m
m m 1
x 0 1( m x ) m 1
m x
m 2
x
2m
x
( )
m 1 2m m 1
m x m x
77.
2
nRT
V nb
an2 .
V
2 2
dP ( V nb) 0 ( nRT )(1) V (0) ( an )(2 V ) 2
nRT 2an
dV
2 2 2 2 3
V nb V V nb V
P We are holding T constant, and a, b, n, R are also constant so their derivatives are zero
( ) ( ) ( )
78.
A( q)
km
q
cm
hq
2
1
( km) q cm
h
q
2
dA
dq
2
( km)
q
h km h
2
2
q 2
2
d A
3 2
2
2( km)
q
km
3
dt
q
3.4 THE DERIVATIVE AS A RATE OF CHANGE
1.
2.
2
s t 3t 2, 0 t 2
(a) displacement s s(2) s(0) 0 m 2 m 2 m, v
s 2
av
1 m/sec
t 2
2
(b) v
ds
2t 3 | v (0)| | 3| 3 m/sec and | v(2)| 1 m/sec; a
d s
2 a(0)
dt
2
dt
2
a(2) 2 m/sec
(c) v 0 2t 3 0 t 3 . v is negative in the interval 0 t 3 and v is positive when 3 2
2
2
body changes direction at t 3
2 .
2
s 6 t t , 0 t 6
(a) displacement s s(6) s(0) 0 m,
s 0
ds
dt
v av t
6
0 m/ sec
(b) v 6 2t | v (0)| |6| 6 m/ sec and | v(6)| | 6| 6 m/ sec; a
2
2
d s
2
dt
2
2 m/sec and
t 2 the
2 a(0) 2 m/ sec and
a(6) 2 m/ sec
(c) v 0 6 2t 0 t 3. v is positive in the interval 0 t 3 and v is negative when 3 t 6 the
body changes direction at t 3.
2
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