derivatives - How should we interpret $frac {d}{dt}$?

I've been using derivatives and integrals mechanically for years without really questioning the symbols. I recently watched some YouTube videos and came to understand that: $$\frac {dx}{dt}$$ basically means, for some function, $f(t)=x$, an infinitesimal change in $t$, or $dt$, results in an infinitesimal change in $x$, or $dx$. The ratio of those two numbers is the derivative, or the instantaneous tangent line of $f(t)$ at $t$. So far, so good.

So could someone explain how to interpret this: $$\frac {d}{dt}$$ I get that the bottom part is an infinitesimal change in $t$, but what is the top part? And how should I read an expression like $$\frac {d^2x}{dt^2}$$ My main confusion is the $d$ part seems to have an existence on it's own without the dimension.