I'm learning about differentials on my own time. In the lecture the instructor mentioned that treating $dx$ as number was not a good idea. I'm not very clear why. Let's talk about it in concrete terms.
Say we have a square with the side $x$. The area of the square is $$A(x)=x^2$$ $$\frac{dA}{dx} = 2x$$ Here is what some might find controversial: $$dx*\frac{dA}{dx} = 2x *dx$$ $$\require{cancel}\cancel{dx}*\frac{dA}{\cancel{dx}} = 2x *dx$$ $$dA=2x*dx$$
From what I remember from my calculus class $dx$ and $dA$ are really small values, meaning I can multiply, divide etc. Can someone explain why what I am doing could be considered contoversial?