The uniqueness of differential polynomials of meromorphic functions ${f^n}f'$ and ${g^n}g'$ IM sharing a polynomial $P(z)$ was studied. If $n > 22$ and the degree of $P(z)$ is not more than $n$, then $f(z) = tg(z)$, or $f(z) = {\lambda _1}{{\rm{e}}^{{\rm{\lambda}} \int {{{P}}({{z}}){{{\rm d}z}}} }}$, $g(z) = {\lambda _2}{{\rm{e}}^{ - {\rm{\lambda}} \int {{{P}}({{z}})} {\rm d}z}}$, where $t$, ${\lambda _1},{\lambda _2}$, $\lambda $ are constants.