Why is ($dx)^2$ tiny quantity of $x^2$ and not of $x?$

Question

When we say dx we are referring to a tiny quantity of $ x$ . And from literature the $(dx)^2$ is a tiny quantity of $x^2$
This is the part I am not clear about.
A minute is $\frac{1}{60}$ of an hour so we could denote $\frac{1}{60}$ as $dx$ and the hour as $x$
Now a second $(\frac{1}{60})^2$ is a tiny quantity of a minute and therefore a tiny quantity of a tiny quantity of an hour.
But I already said that hour is $x$.
So how can we also say that it is a tiny quantity of $x^2$? (which is I guess $hour^2$?)

Answer 1

The notation $\mathrm{d}x$ should not be attempted to be taken too literally. While it can be made precise it's usually considered a relic from time when things were not as precisely defined as they are these days. You will run in all kinds of troubles if you try to naively assign it a deeper meaning.

I've never seen $\mathrm{d}x^2$ though.

Answer 2

We don't have good intuition about what time-squared means, so maybe you should use length, since length squared is area. If you had a square which was $x$-by-$x$ feet and then increased both sides by $dx$, then the area of your new square is $x^2 +2x \; dx+ (dx)^2$. The change in the area is two skinny $x$-by-$dx$ strips and then a little corner piece of area $(dx)^2.$ So it's not really the change in $x^2$, but it's a measure of how much the change in $x^2$ is not linear.

Or you could just consider the skinny strip. Then $(dx)^2$ is (sort of) the change from $x$-by-$dx$ to $x+dx$-by-$dx$.

As others have said, it takes a decent sized pile of mathematical machinery to really define and deal with differentials rigorously.