First some data you need to have in order to calculate the DA
accurately:
Temperature at sea level (ISA): T0ISA = 273,15 K + 15 K = 288,15 K
Temperature (let's say ISA+1): T = T0ISA + 1K = 289,15 K
Static pressure at sea level (ISA): p0ISA = 1013,25 hPa = 101325 Pa
Static pressure (let's say it's the same as ISA since you are only interested in change of DA in light of temperature): p = p0ISA = 101325 Pa
Temperature gradient: a = 0.0065 K/km (notice: kilometers!!)
Gas constant: R = 287 J/(kg * K)
Gravity acceleration: g = 9,81 m/s^2
Air density at sea level (ISA): Rho0 = 1.225 kg/m^3
First, we are interested in effect of temperature increase on air density:
Rho = p / (R * T) = 1.2207 kg/m^3
Now we can calculate the density altitude with the following formula:
DA = (T0ISA / a) * (1 - (Rho / Rho0))^((R * a) / (g - R * a)) = 36.258 m
All we have to do now is to convert DA from meters to feet:
DAf = DA * 3.28 ft/m = 118.928 ft
Very close to the 118.8 ft/K that is mentioned throughout the ATPL books. Nice to see that at least few people care to really understand what is behind rules of thumb and ATPL theory, not just learn the answers to the questions
On
Wikipedia you can find another formula, which gave me the result of 117.794 ft/K and probably suffers from rounding errors.