Section 4.5
Differentiate from first principles:
- 2 x
2
1 + 13 x 1 + 14 19.x
4
- 22 x
2
21.x
322
22.e
−xSection 4.6
Differentiate by rule:
23.x
3
24.x
524
25.x
123
- 12 x
3
- 11 − 12 x 1 + 13 x
2
1 − 14 x
3
1 + 151 sin 1 x 1 − 161 cos 1 x 1 + 17 e
x1 − 181 ln 1 x
28.The virial equation of state of a gas at low pressure is. Find
at constant Tand n (assume Bis also constant).
Products and quotients
Differentiate
29.(1 1 − 14 x
2
) 1 cos 1 x 30.(2 1 + 13 x)e
x31.e
x1 cos 1 x 32.x 1 ln 1 x
33.(1 1 + 12 x 1 + 13 x
2
) 2 (3 1 + 1 x
3
) 34.(1 1 − 14 x
2
) 2 sin 1 x 35.cos 1 x 2 sin 1 x 36.(1n 1 x) 2 x
Chain rule
Differentiate
37.(1 1 + 1 x)
5
41.sin 14 x 42.e
− 2 x- 44 .ln(2x
2
1 − 13 x 1 + 1 1)
45.cos(2x
2
1 − 13 x 1 + 1 1) 46.e
sin1
x47.ln(cos 1 x) 48.
- 50.ln(sin 12 x 1 + 1 sin
2
1 x) 51. 3 x
2
(2 1 + 1 x)
122
52.sin 1 x 1 cos 12 x
53.tan 14 x 1 cos
2
12 x 54. 55.
Inverse functions
56.Ifx 1 = 12 y
2
1 − 13 y 1 + 1 1, find.
Find at constant Tand nfor the following equations of state (assume that B,a andb
are constants).
dV
dp
dy
dx
3
2
2
212
x
()x
+xe
22 3x2+
ln
2
3
−
x
x
e
−+cos(x ) 322e
231 xx2−+
3
231
212
()xx−−
1
3
2
−x
2
2
+x
dp
dV
pV nRT
nB
V
=−
1
lim ln( ) ln( )
x
xx
→
−− +
∞
lim (ln ln ) 432
x
xx
→
−
0
2
lim
x
e
x
x→
−
02
1
lim
x
x
x
x
x
→
−
+−
0
4
1
2
1
22
2
lim
x
x
x
→
−
∞2
1
1
lim
x
x
x
→
−
−
∞
1
1
2
lim
x
x
→ x
∞
1
3
lim
x
x
x
→
−
−
1
1
1
2
4.13 Exercises 123