I came across a formula that is supposed to be the error in Numerical Differentiation when interpolating a function $f(x)$ using the points $x_0,x_1,\dots,x_n$ :
$$e^{'}(x)= \frac{ψ(x)}{(n+2)!}*f^{(n+2)}(ξ_1)\space + \space \frac{ψ^{'}(x)}{(n+1)!}*f^{(n+1)}(ξ_2) $$ where :
- $ξ_1,ξ_2 \in (x_0,x_n)$
- $ψ(x) = \prod_{i=0}^{i=n} (x-x_i)$
- $e(x)=\frac{\prod_{i=0}^{i=n} (x-x_i)}{(n+1)!}*f^{(n+1)}(ξ)\space, ξ \in (x_0,x_n)$
I fully understand how the error formula $e(x)$ is calculated in Polynomial Interpolation and the $(n+1)!$ term is in it as $ψ(x)$ is a polynomial of $n+1$ degree but I struggle to understand why the $(n+2)!$ term is in $e^{'}(x).$