The best, by coefficients, weighted quadrature formulas of the form $\sum\limits_{k=1}^n \sum\limits_{i=0}^{r-1} A_{ki} f^{(i)}_{jk}$ for some classes of differentiable functions

For the quadrature formula (with non-negative, integrable on $[0,1]$ function) that is defined by the values of the function and its derivatives of up to and including $(r-1)$-th order, we find the form of the best coefficients $A^0_{ki}$ ($k = \overline{1, n}$, $i = \overline{0, r-1}$) for fixed nodes $\gamma_k$ ($k = \overline{1, n}$) and we give the sharp estimate of the remainder of this formula on the classes $W^r_p$, $r = 1, 2, \ldots$, $1 \leqslant p \leqslant \infty$.