Is writing $dx^2$ same as writing $d(x^2)$ in calculus

Question

When we write $dx^2$, do we actually mean $$d(x^2)$$ (the change in respect to $x^2$), or $$(dx)^2$$ ($dx$ squared)?

Answer 1

The usual notation in Stieltjes integrals for differentiable $f(x)$ is $$ df(x) = f'(x) dx $$ so in your case $$ d\left(x^2\right) = 2xdx $$