  Over a field of characteristic $p>2,$ the first cohomology of the
3-dimensional simple Lie algebra $\frak{sl}(2)$ with coefficients in all simple
modules is determined, which implies Whitehead's first lemma is not true in
prime characteristic. As applications, derivations and 2-local derivations of
$\frak{sl}(2)$ on any simple module are also be characterized.
