TITLE: Why do we write second derivatives like $\frac{d^2x}{dt^2}$
QUESTION [5 upvotes]: Why do we write the second derivative of $x$ with respect to $t$ as $\frac{d^2x}{dt^2}$?
It's never been explained to me, and I've never found a particularly good explanation. What's up with this weird derivative notation?

REPLY [5 votes]: One derivation goes as follows,
$$\frac{d}{dt}\left(\frac{d}{dt}(x)\right) \to \frac{d}{dt}\left(\frac{dx}{dt}\right) \to \frac{d\left(\frac{dx}{dt}\right)}{dt} \to \frac{ddx}{dtdt} \to \frac{d^2x}{(dt)^2} \to \frac{d^2x}{dt^2}.$$