What is the meaning of $dy=dx^2$?

Question

When I read the mathematical analysis ,I think if the differential is $dy=Adx^2$ $A$ is a function about x, what will happen? Maybe, it is not proper defined ,but I think the "function" meet $dy=A(dx)^2$ will has some fractal structure.

Answer 1

$\mathrm dy = \mathrm dx^2 \\ \Rightarrow \mathrm dy = 2x\cdot\mathrm dx \\ \Rightarrow \frac{\mathrm dy}{\mathrm dx} = 2x$

Usually,

$d\left(f(x)\right) = f'(x) \cdot \mathrm dx$

Answer 2

Hints:

$dy=Adx^2=(2xA)dx$. In fact, $df(x)=f'(x)dx$.

Answer 3

These are differentials. They're handy for linear approximations and errors. Essentially, it's really just another way of representing a derivative.

Since everybody is differentiating, I figure I'll integrate instead.

$$\int dy = \int A \ d(x^2).$$ I take it that $A$ is a constant, so $y = A x^2 + C$. If $A(x)$ were a function of $x$, we would make the substitution $u=x^2$, and have $$y = \int A(\sqrt{u}) \ du,$$ which is kind of awkward.