How to interpret operator form

Question

I was looking over a document covering various methods to solve linear DEs containing variable coefficients. The document mentions something called "operator form" and proceeds to abbreviate the operator $\frac{d}{dt}$ with the symbol $D$, and then perform operations on it. It goes through some steps, like changing variables, (none of which is important to this question) but we eventually reach this statement: $$\frac{d}{dx}=e^{-t}D$$ This supposedly implies that: $$\frac{d^{2}}{dx^{2}}=e^{-t}De^{-t}D=e^{-2t}(D-1)D$$ How am I supposed to interpret $e^{-t}De^{-t}D$, i.e., order of operations? Do I read it from left to right? Is this equivalent to $(e^{-t}D)(e^{-t}D)$ or $((e^{-t}D)e^{-t})\cdot D$? What is $(D-1)$?