## Empty one year ago Is the product rule broken?

1. Empty

Ok obviously not, but something weird happened when I was considering a change of area of a square by an infinitesimal amount: |dw:1438695359737:dw| 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

2. phi

dx dy is "second order" and goes to zero somewhere in all of this there is a limit, that causes that term to drop out.

3. Empty

Ahh ok I guess that makes sense.

4. Zarkon

instead of $dA=ydx+xdy+dxdy$ you should probably write $\Delta A=y\Delta x+x\Delta y+\Delta x\Delta y$ then $\frac{\Delta A}{\Delta x}=\frac{y\Delta x}{\Delta x}+\frac{x\Delta y}{\Delta x}+\frac{\Delta x\Delta y}{\Delta x}$ $\frac{\Delta A}{\Delta x}=y+x\frac{\Delta y}{\Delta x}+\Delta y$ letting $$\Delta x\to0$$ and therefore $$\Delta y\to0$$ $\frac{dA}{dx}=y+x\frac{dy}{dx}+0$ $dA=ydx+xdy$