I am reading this document.
In this article after defining strong derivative Knuth goes on to calculate derivative of $x^n$. There he uses definition of strong derivative to expand $(x+\epsilon)^{n+1}$ as follows,
$(x+\epsilon)^{n+1} = (x+\epsilon)[x^n + d_n(x)\epsilon + \mathcal{O}(\epsilon^2)]$
when I expand the right side I get
$x^{n+1} + (x d_n(x) + \ x^n)\epsilon + \color{red}{x \ \mathcal{O}(\epsilon^2) + \epsilon^2 \ d_n(x) + \epsilon \ \mathcal{O}(\epsilon^2)}$
But in Knuths calculations the red part is just $\mathcal{O}(\epsilon^2)$.
Question is how? I don't know how to work this out in detail.