  Polynomial approximation is studied in the Sobolev space
$W_p^r(w_{\alpha,\beta})$ that consists of functions whose $r$-th derivatives
are in weighted $L^p$ space with the Jacobi weight function $w_{\alpha,\beta}$.
This requires simultaneous approximation of a function and its consecutive
derivatives up to $s$-th order with $s \le r$. We provide sharp error estimates
given in terms of $E_n(f^{(r)})_{L^p(w_{\alpha,\beta})}$, the error of best
approximation to $f^{(r)}$ by polynomials in $L^p(w_{\alpha,\beta})$, and an
explicit construction of the polynomials that approximate simultaneously with
the sharp error estimates.
