I have the following expression
$$\frac{b}{(b+x)^2}$$
where $b>0$. This function seems to be decreasing in $x$ with (at least) square of $x$. However, I also see a different expression of the previous function, where $S=b+x$, then
$$\frac{S-x}{S^2}=\frac{1}{S}-\frac{x}{S^2}$$
Which does not seem to be decreasing in $x$ with the square of $x$. I'm sure I am failing to see something, any suggestions?
You're missing that $S$ increases with $x$. So what you you have is the difference between two terms, both decreasing with $x$. That by itself doesn't necessarily mean that the difference decreases with $x^2$ (see the expression $\frac1x - \frac1{2x}$, for instance), but in this case the two fractions happen to be tuned to one another in such a way that they do.