  D. Leviatan has investigated the behavior of the higher order derivatives of
approximation polynomials of the differentiable function $f$ on $[-1,1]$.
Especially, when $P_n$ is the best approximation of $f$, he estimates the
differences $\|f^{(k)}-P_n^{(k)}\|_{L_\infty([-1,1])}$, $k=0,1,2,...$. In this
paper, we give the analogies for them with respect to the differentiable
functions on $\mathbb{R}$, and we apply the result to the monotone
approximation.
