Question

Is the product rule broken?

Answer 1

Ok obviously not, but something weird happened when I was considering a change of area of a square by an infinitesimal amount:

So infinitesimally changing the area of the square is that extra area there \(dA\) but if you add it up by considering the little changes in x and y separately you get that

Geometrically two slender rectangles and a tiny corner:
\(dA=ydx+xdy+dxdy\)

But if you use the product rule you get:

\(dA=d(xy)=xdy+ydx\)

So where did the dxdy term go? Actually isn't \(dxdy=dA\)? I feel like I've confused myself here haha