From here:
Why isn't the $\displaystyle (-x)^n (1/x)^n$ product simplified to $\displaystyle (-1)^n$?
Thanks in advance for any help.
Because it's an operator not just an algebraic expression. To demonstrate let $\displaystyle n = 2$ (and $\displaystyle f(x) = \frac{\sin x}{x}$ ) so
$\displaystyle
(-x)^2 \left(\frac{1}{x} \frac{d}{dx} \right)^2 f(x) = x^2 \frac{1}{x} \frac{d}{dx} \left(\frac{1}{x} \frac{df}{dx} \right) = \frac{d^2f}{d x^2} - \frac{1}{x} \frac{df}{dx}.
$