I am having difficulties understanding how this operator is multiplied out. I have the answer, but do not know why (see below) it is what it is. Imagine there is a carat (^) above the "A" for correct notation.
Write out the operator $(A)^2$ for
$$A = (d/dx + x)$$
This link is the solution to the problem.
What I am confused about is where did $f(x) \cdot dx/dx$ come from? Also, why does $d/dx \cdot xf(x)$ turn into $x \cdot[df(x)/dx]$, not $[d(f(x) \cdot x)/dx]$? Please explain. Thanks!
Your confusion stems from the application of the product rule, which is generally: $$\frac{d}{dx}(u\cdot v) = \frac{du}{dx} \cdot v + u \cdot \frac{dv}{dx}$$
In your case it is: $$\frac{d}{dx}\Big(x\cdot f(x)\Big) = \frac{dx}{dx} \cdot f(x) + x \cdot \frac{df(x)}{dx}$$