Bank Angle = arctan(v^2/11.26r)
If you are interested in rate of change, the units don't matter.
The formula in SI units is (Bank Angle) = arctan (v^2 / g r)
or r = g / v^2 tan BA
Where A = bank angle and 1/tanA = cotA, and assuming that v is constant:
dr/dA = d/dA (g/v^2tanA)
= g/v^2 dcotA / dA
= g/v^2 (-cosec^2 A)
= g/v^2 (-1/sin^2 A)
If you plot radius (vertical axis) against bank angle (horizontal), at about 100kTAS, the gradient of the line at each value of bank angle is as follows. The "-" just means that as bank increases, radius decreases. The smaller the number, the less the radius will change for each 1° increase in bank.
5° -0.53
10 -0.13
15 -0.06
20 -0.03
25 -0.022
30 -0.016
35 -0.012
40 -0.010
Been years since I've done that sort of thing. Someone more engineering than I have, show me where I went wrong
If you want absolute values, the above formula uses radius in meters, v in m/s, g in m/s/s (9.81). Then you can convert into whatever units your bugsmasher is calibrated for...