124 Chapter 4Differentiation
- 58.p(V 1 − 1 nb) 1 − 1 nRT 1 = 10 59.
Differentiate
60.sin
− 1
12 x 61.tan
− 1
1 x
2
- 63.sinh
− 1
12 x 64.tanh
− 1
1 x
2
Section 4.7
Find
65.x
2
1 + 1 y
2
1 = 14 66.y
3
1 + 13 x 1 + 1 x
2
1 − 111 = 10 67.x 1 = 1 y 1 ln 1 xy
Section 4.8
Differentiate:
- 71.sin
122
1 x 1 cos
3
(x
2
1 + 1 1)tan
123
12 x
72.Show that the equations
are equivalent expressions of the Clausius–Clapeyron equation.
73.The decomposition of dinitrogen pentoxide in tetrachloromethane atT 1 = 1 45°Chas
stoichiometry:
and obeys first-order kinetics. From the volumes of oxygen liberated after various times t,
the following concentrations of N
2
O
5
were obtained:
x 1 = 1 [N
2
O
5
] 2 mol 1 dm
− 3
2.33 1.91 1.36 1.11 0.72 0.55
t 2 s 0.0 319 867 1196 1877 2315
Plot a graph of ln xagainstt 2 sand determine the rate constant.
Section 4.9
74.Find all the nonzero derivatives of the functiony 1 = 13 x
5
1 + 14 x
4
1 − 13 x
3
1 + 1 x
2
1 − 12 x 1 + 11.
75.Find for the functiony 1 = 1 ln x.
76.Find a general formula for the nth derivative ofe
3 x.
77.Find a general formula for the nth derivative of cos 12 x.
Section 4.10
Find the maximum and minimum values and the points of inflection of the following
functions. In each case, sketch the graph and show the positions of these points.
78.y 1 = 1 x
2
1 − 13 x 1 + 12 79.y 1 = 1 x
3
1 − 17 x
2
1 + 115 x 1 − 19 80.y 1 = 14 x
3
1 + 16 x
2
1 + 13
81.y 1 = 1 xe
−x(see Figure 3.20)
dy
dx
dy
dx
dy
dx
dy
dx
,,,
2
2
3
3
4
4
NO NO O
25 2
1
2 2
→+ 2
dp
dT
H
RT
dp
dT
pH
RT
ln
=
∆
=
∆
vap vapand
22
()()
()( )
11
213 21
212
213
+−
++−
xx
xxx
3
4
13
−
x
x
y
y
xy x
222
2
+− + += 320
dy
dx
:
cos
−
−
1
1
1
x
x
p
na
V
- Vnb nRT
−=
2
2
pV nRT ()
nB
V
=+
1