Let N be a prime near-ring with center Z. The purpose of this paper is to study derivations on N. We show two main results: (1) Let N be 2-torsion-free, and let
D
1
{D_1}
and
D
2
{D_2}
be derivations on N such that
D
1
D
2
{D_1}{D_2}
is also a derivation. Then either
D
1
{D_1}
or
D
2
{D_2}
is zero if and only if
[
D
1
(
x
)
,
D
2
(
y
)
]
=
0
[{D_1}(x),{D_2}(y)] = 0
for all
x
,
y
∈
N
x,y \in N
. (2) Let n be an integer
≥
1
\geq 1
, N be n!-torsion-free, and D a derivation with
D
n
(
N
)
=
{
0
}
{D^n}(N) = \{ 0\}
. Then
D
(
Z
)
=
{
0
}
D(Z) = \{ 0\}
.