Some estimates for simultaneous polynomial approximation of a function and its derivatives are obtained. These estimates are exact in a certain sense. In particular, the following result is derived as a corollary: For \(f\in C^r[-1,1]\), \(m\in\mathbb{N}\), and any \(n\geq\max\{m+ r-1,2r+1\}\), an algebraic polynomial \(P_n\) of degree \(\leq n\) exists that satisfies \[ |f^{(k)}(x)- P^{(k)}_n(f,x)|\leq C(r,m) \Gamma_{nrmk}(x)^{r- k}\omega^m(f^{(r)}, \Gamma_{nrmk}(x)), \] for \(0\leq k\leq r\) and \(x\in[-1,1]\), where \(\omega^\nu(f^{(k)},\delta)\) denotes the usual \(\nu\)th modulus of smoothness of \(f^{(k)}\), and \[ \begin{multlined} \Gamma_{nrmk}:=\\ \begin{cases} n^{-1}\sqrt{1- x^2},\quad &\text{if }x\in[-1+ n^{-2},1- n^{-2}]\\ (1- x^2)^{(r- k+1)/(r- k+m)}({1\over n^{-2}})^{(m- 1)/(r- k+m)},\quad &\text{if }x\in [-1,-1+ n^{-2}]\cup [1- n^{-2},1]\end{cases}\end{multlined} \] Moreover, for no \(0\leq k\leq r\) can \((1- x^2)^{(r- k+1)/(r- k+m)}(1/n^{-2})^{(m- 1)/(r- k+m)}\) be replaced by \((1- x^2)^{\alpha_k} n^{2\alpha_k-2}\), with \(\alpha_k>(r- k+1)/(r- k+m)\).