Help understanding proof for product rule (specifically distribution of ${\Delta(uv)\over\Delta x}$)

The proof starts like this:

${d\over dx}(uv) = \lim \limits_{\Delta x \to 0}{\Delta(uv)\over\Delta x} = \lim \limits_{\Delta x \to 0} \left(u{\Delta v \over \Delta x} + v{\Delta u \over \Delta x} +\Delta u{\Delta v \over \Delta x}\right)$

The three terms in the parentheses, how is it that we get these from ${\Delta(uv)\over\Delta x}$? I'm not that familiar with this syntax (from Stewart's Calculus Early Transcendentals, 7th edition) as in class we don't use the Delta symbols (yet anyway). Why doesn't $\Delta(uv)$ distribute to just $\Delta uv$ like $3(ab) => 3ab$?

$\Delta(uv)$ means the change in the product $uv$.
\begin{align*}\Delta(uv) & = (u+\Delta u)(v+\Delta v) - uv \\ & = uv + u\Delta v + v\Delta u + \Delta u \Delta v -uv \\ & = u\Delta v + v\Delta u + \Delta u \Delta v\end{align*}
Then $\dfrac{\Delta(uv)}{\Delta x} = \dfrac{u\Delta v + v\Delta u + \Delta u \Delta v}{\Delta x} = u\dfrac{\Delta v}{\Delta x} + v\dfrac{\Delta u}{\Delta x} + \Delta u \dfrac{\Delta v}{\Delta x}$