At time 1:06 of this video by minutephysics, there is a geometric representation of the product rule:
However, I don't understand how the sums of the areas of those thin strips represent $d(u\cdot v)$. The only way I can see it is that $d(u\cdot v)$ is a small change in the area of the square, and those thin strips do represent that; however, I'm not sure if this is correct and if it is, how formal of a proof is this? Thanks!
I use the picture of the rectangle in my own teaching (without the differential notation) and show it to grad students who are starting their teaching careers. It is far superior to the usual tricky addition-of-$0$ argument found in most textbooks.
Here is the argument in greater detail:
\begin{align*} \frac{\Delta(uv)}{\Delta x} &= \frac{(u+\Delta u)(v+\Delta v) - uv}{\Delta x} \\ &= \frac{u\Delta v + v\Delta u + \Delta u\Delta v}{\Delta x} = u \frac{\Delta v}{\Delta x} + v \frac{\Delta u}{\Delta x} + \Delta u\cdot\frac{\Delta v}{\Delta x}\,. \end{align*} Taking $\lim\limits_{\Delta x\to 0}$ gives the product rule.
This can all be written out with the usual $f(x+h)g(x+h)$ notation, if so desired.
By the way, this same picture can be used to give a more motivated proof of the product theorem for limits, as well.