Like operators? If so, you might say something like...
Let:
#color(green)(hatD = d/(dx)[f(x)])#
and
#hatD^((n)) = d/(dx)[d/(dx)[d/(dx)[cdotsd/(dx)[f(x)]]]]#
Then:
#color(blue)(hatD^((3))[f(x)]) = d/(dx)[d/(dx)[d/(dx)[f(x)]]]#
#= d/(dx)[d/(dx)[f'(x)]]#
#= d/(dx)[f''(x)]#
#= color(blue)(f'''(x))#
...by evaluating from the inside out, right to left.
Or perhaps, for linear momentum (and this is an actual operator, in Physical Chemistry):
Let:
#color(green)(hatp = -(ih)/(2pi)d/(dx))#
Thus:
#color(blue)(hatpf(x)) = -(ih)/(2pi)d/(dx)[f(x)] = color(blue)(-(ih)/(2pi) f'(x))#
Since operators operate from right to left here you would take the derivative first, and then multiply by #-(ih)/(2pi)#.
If you said...
Let:
#color(green)(hat(LN) = ln[f(x)])#
and
#hat(LN)^((n)) = ln[ln[...ln[f(x)]]]#
Then:
#color(blue)(hat(LN)^((2))) = ln[ln[f(x)]] = ln[ln(f(x))] = color(blue)(ln(ln(f(x))))#
(I said #hat(LN)# instead of #hatL# because #hatL# is the angular momentum operator in the x, y, or z direction in Physical Chemistry)
