Name and derivation of the approximation $\frac{1+x}{1+y} - 1 \approx x-y$? I am wondering if there is a name and way to derive the following approximation:
$$\frac{1+x}{1+y} - 1 \approx x-y$$
I'm essentially interested in how to refer to this.

I would call this a Taylor approximation. When $|y|\lt1$,
$$
\begin{align}
\frac{1+x}{1+y}-1
&=-1+(1+x)\left(1-y+y^2-y^3+\dots\right)\\
&=x-y-xy+y^2+xy^2-y^3-xy^3+\dots\\[6pt]
&=x-y+O\!\left(\max(|x|,|y|)^2\right)
\end{align}
$$
