I know when interpolating a function $f(x)$ by a degree $n$ polynomial $P(x)$, the error is
$$f(x) - P(x) = \frac{f^{n+1}(\xi)}{(n+1)!}\prod_{i=0}^n(x-x_i)$$
What does the right side hand look like for the error of the derivatives?
$$f^{(n)}(x) - P^{(n)}(x) = \text{???}$$
Also the product factor is minimized by the Chebyshev nodes. How about when the points $x_0 = -1$ and $x_n = 1$ are added as constrains? I found the first ones:
$$\{-1, 1\}\quad\{-1, 0, 1\}\quad\left\{-1,1-\sqrt{2},\sqrt{2}-1,1\right\}\\\left\{-1,\left(1-\sqrt{5}\right)/2,0,\left(\sqrt{5}-1\right)/2,1\right\}\\\left\{-1,-\sqrt{2 \left(2-\sqrt{3}\right)},-\sqrt{7-4 \sqrt{3}},\sqrt{7-4 \sqrt{3}},\sqrt{4-2 \sqrt{3}},1\right\}$$
But I fail to see a pattern. Is there any references?