Product rule intuition

If you think of $fg$ as giving the area of a rectangle with side lengths $f$ and $g$ (assuming $f,g$ are non-negative, otherwise just flip their signs), then if you change $x$ slightly then one side of the rectangle changes by an amount proportional to $df/dx$ and the other side of the rectangle changes by an amount proportional to $dg/dx$. So you get two little strips of changed area, which have total area proportional to $f'g + g'f$. The only question is whether the overlap of the two little strips matters, and it turns out that it doesn't because the overlap of the two strips has infinitesimal area which is a product of two infinitesimals (really really small), whereas the two strips have infinitesimal area which is a product of only one infinitesimal each (really small, but much larger than the overlap).

I like to think of it in terms of units and symmetry. We know that, unit-wise, the false formula

$$\frac{d}{dt} (f g) = \frac{df}{dt} \frac{dg}{dt}$$

cannot hold, because, if say $f$ and $g$ each represent distance and $t$ time, the LHS has units of distance^2/time, while the RHS has units distance^2/time^2.

Thus, the product rule for derivatives, in order to keep units, must be of the form $f dg/dt$ or vice-versa. But because of the symmetry of the original form, the product rule should also be symmetric, i.e. $f dg/dt + g df/dt$. As there is no other symmetric form that keeps the units, this should be the product rule, at least intuitively.

$$(x+\delta x)(y+\delta y)=xy + (x\delta y+y\delta x) + O(\delta^2)$$