First timer here.
I would like to know the correct ways of writing and interpreting derivatives written using leibniz notation. I have a sample of 8 symbols below. For each one of them I would like to know:

* whether it is incorrect notation
* if it is not incorrect whether there is a better way to write the same
* what function is the derivative operator acting on

$$\begin{array}{rrrr}
\frac{\mathrm{d}x^2}{\mathrm{d}x}, &
\frac{\mathrm{d}kx^2}{\mathrm{d}x}, &
\frac{\mathrm{d}x^2+x^3}{\mathrm{d}x}, &
\frac{\mathrm{d}\left(x^2+x^3\right)}{\mathrm{d}x},\\
\frac{\mathrm{d}}{\mathrm{d}x}x^2, &
\frac{\mathrm{d}}{\mathrm{d}x}kx^2, &
\frac{\mathrm{d}}{\mathrm{d}x}x^2+x^3, &
\frac{\mathrm{d}}{\mathrm{d}x}\left(x^2+x^3\right)
\end{array}
$$