How do I calculate density altitude by hand?
Density altitude is a value that directly correlates aircraft performance to atmospheric conditions.


  Density altitude is pressure altitude corrected for non-standard temperature.


Almost all temperatures in aviation is Celsius including this formula.



You will notice that all aircraft performance calculations require pressure altitude and temperature and in effect is calculating density altitude.

To calculate pressure altitude you will calculate 
$$ \left( 29.92 - \frac{\mathrm{current~altimeter~setting}}{\mathrm{inHg}} \right) \cdot 1000 \, \mathrm{ft} + \mathrm{current~field~elevation}$$

For example if the current altimeter is $30.12 \, \mathrm{inHg}$ and you are $ 6,400 \, \mathrm{ft} $ MSL, the pressure altitude is
$$ \left( 29.92 - 30.12 \right) \cdot 1000 \, \mathrm{ft} + 6,400 \, \mathrm{ft} = 6,200 \, \mathrm{ft}$$
pressure altitude.



Your formula to calculate density altitude is correct but is never required to be known to calculate aircraft performance as the "calculation" is being done in the background of the performance charts.

To calculate density altitude, you must first calculate pressure altitude and then apply it to your formula.

If you are at 10°C at $6400 \, \mathrm{ft}$ MSL with an altimeter setting of $30.12 \, \mathrm{inHg}$:
$$(10-2.6) \cdot 120 \, \mathrm{ft} + 6,200 \, \mathrm{ft} = 7,088 \, \mathrm{ft}$$ density altitude



I have never calculated density altitude once I got into the air.  I have calculated density altitude while determining if its a good idea to takeoff.  In higher elevation airports on a warm day, the airplane may not get out of ground effect due to the density altitude.

The formula for density altitude above is correct for any altitude, albeit a more simplistic formula.  Humidity also has an effect of density altitude and there are formulas that take humidity into account when calculating density altitude.