Where is misunderstanding dA = dx * dy?

Question

Using infinitesimals from $ A(x, y) = x * y $ I have $ dA = A_x * dx + A_y * dy $ solving it for $ dA $, I have $ dA = y * dx + x * dy $ which is a mistake. Where in my thinking way is that mistake? I have tried with $ A(x, y) = const $ but it is also wrong.

Answer 1

In a double integral $\iint_D f(x,y) \, dA = \iint_D f(x,y) \, dxdy$, the symbol $dA$ is just shorthand for the area element $dxdy$, “a little piece of area”.

That's something completely different from the differential $dA=y\,dx+x\,dy$ of the function $A(x,y)=xy$!

So there's no contradiction, just the same symbol used for two different things.