In this article, we deal with the uniqueness problems of meromorphic functions
concerning differential polynomials and prove the following result: Let $f$ and
$g$ be two nonconstant meromorphic functions and let $n(\geq 14)$ be an integer
such that $n+1$ is not divisible by $3$. If $f^{n}(f^{3}-1)f'$ and
$g^{n}(g^{3}-1)g'$ share $(1,2)$ or $``(1,2)"$, then $f\equiv g$. If
$\overline{E}_{4)}(1,f^{n}(f^{3}-1)f')=\overline{E}_{4)}(1,g^{n}(g^{3}-1)g')$
and $E_{2)}(1,f^{n}(f^{3}-1)f')=E_{2)}(1,g^{n}(g^{3}-1)g')$, then $f\equiv
g$.