Big-O notation in division Let $r(x)=\frac{p(x)}{q(x)}$. Expending $p$ and $q$ around 0 gives
$$
\frac{p_0+p'_0x+\mathcal{O}(x^2)}{q_0+q'_0x+\mathcal{O}(x^2)}.
$$
Now the claim is that the above expression is equal to
$$
\frac{p_0+p'_0x}{q_0+q'_0x}+\mathcal{O}(x^2).
$$

Try to evaluating the difference:
$\displaystyle \frac{p_0+p'_0x+\mathcal{O}(x^2)}{q_0+q'_0x+\mathcal{O}(x^2)}-\frac{p_0+p'_0x}{q_0+q'_0x}$
and recall that $p\mathcal{O}(x^2)=\mathcal{O}(x^2)$ and $x\mathcal{O}(x^2)=\mathcal{O}(x^2)$.
