Normalized derivative of a polynomial ($D_{j}P$)

The $j$th normalized derivative isIf $P(X)={\sum }_r{a}_r{X}^r$ has integer coefficients, then $D_{j}P(X)={\sum }_{r\ge j}\left(\genfrac{}{}{0ex}{}{r}{j}\right)a_r{X}^{r-j}$ also has integer coefficients. For $\mathrm{deg}P\le n$, its naive polynomial height is at most $2^{n}H(P)$.