75. We take the derivative with respect to x of both sides of Eq. 34-11:
∂
∂x
∂E
∂x
=
∂
2
E
∂x
2
=
∂
∂x
−
∂B
∂t
=
−
∂
2
B
∂x∂t
.
Now we differentiate both sides of Eq. 34-18with respect to t:
∂
∂t
−
∂B
∂x
=
−
∂
2
B
∂x∂t
=
∂
∂t
ε
0
µ
0
∂E
∂t
= ε
0
µ
0
∂
2
E
∂t
2
.
Substituting ∂
2
E/∂x
2
=
−∂
2
B/∂x∂t from the first equation above into the second one, we get
ε
0
µ
0
∂
2
E
∂t
2
=
∂
2
E
∂x
2
,
or
∂
2
E
∂t
2
=
1
ε
0
µ
0
∂
2
E
∂x
2
= c
2
∂
2
E
∂x
2
.
Similarly, we differentiate both sides of Eq. 34-11 with respect to t
∂
2
E
∂x∂t
=
−
∂
2
B
∂t
2
,
and differentiate both sides of Eq. 34-18with respect to x
−
∂
2
B
∂x
2
= ε
0
µ
0
∂
2
E
∂x∂t
.
Combining these two equations, we get
∂
2
B
∂t
2
=
1
ε
0
µ
0
∂
2
B
∂x
2
= c
2
∂
2
B
∂x
2
.