Why can $st \left(\frac{\Delta y}{\Delta x}\right)=st \left(\frac{\Delta u}{\Delta x}\right)+st \left(\frac{\Delta v}{\Delta x}\right)$ be written as $\frac{dy}{dx}=\frac{du}{dx}+\frac{dv}{dx}$?
This is taken from Elementary Calculus: an infinitesimal approach. You can see a picture illustrating the origin of my "problem":
Help appreciated, as I cannot see what happened.
If $y$ can be expressed as a function of $x$, say $y = f(x)$, then you have previously defined
$$ f'(x) = \operatorname{st}\left( \frac{\Delta y}{\Delta x} \right) $$
$$ dy = f'(x) dx $$
from which it follows
$$ \frac{dy}{dx} = \operatorname{st}\left( \frac{\Delta y}{\Delta x} \right) $$