If d(dx) = d²x then why not possible dt(dt) = d²t²? Still I don't understand. Please simplify.
This is entirely a matter of notation -- how we choose to write the second derivative. Ultimately, you just have to learn that this is how it is written.
But the motivation is that, using \(\displaystyle \displaystyle \frac {d}{dx}\) to represent the operation of taking the derivative of a function, the second derivative is represented as \(\displaystyle \displaystyle \frac{d}{dx}\left(\frac{dy}{dx}\right)\), and we write that
as if d were a number,
even though it isn't: \(\displaystyle \displaystyle \frac{ddy}{dx dx} = \frac{d^2y}{(dx)^2}= \frac{d^2y}{dx^2}\). We don't bother to use parentheses around the dx because in some sense the d, as a "differential operator", is thought of as tightly bound to what follows it; and we don't distribute and write d
2x
2 because that would suggest we are doing something to x
2, which we are not.
Another way to look at it is in terms of the definition. The derivative is a limit of a fraction, \(\displaystyle \displaystyle\frac {dy}{dx} = lim_{\Delta x\to 0}\frac {\Delta y}{\Delta x}\), so the second derivative is the limit of \(\displaystyle \displaystyle \frac {\Delta \frac {\Delta y}{\Delta x}}{\Delta x}\), and this can
almost be thought of as \(\displaystyle \displaystyle \frac {\Delta(\Delta y)}{(\Delta x)^2}\). Here, again, \(\displaystyle \Delta\) is not a number, but an operator, so distribution is not valid.
But, again, this is simply a notation that reminds us of these relationships, and happens to work well; if you try to analyze it too closely, it falls apart. Just accept it as the way we write these things.
