Note that the definition uses a temperature decrease with increasing altitude, which is the normal temperature profile in the troposphere.

When subtracting to get the difference between the temperatures and altitudes, always subtract in the same subscript order and preserve the negative/positive signs.

Because our definition uses a temperature decrease with increasing altitude, we need to put a negative sign in front of the formula, since a "normal" temperature profile will have a mathematically negative slope and we want this to represent a positive lapse rate.

The lapse rate can describe the temperature variation in the environment, if you take a thermometer to different altitudes in the atmosphere. We can also express a lapse rate for a rising or sinking air parcel, if we keep a thermometer inside the parcel as it moves around. We get a value of 10°C/km for a parcel rising adiabatically (and expanding). This value is more or less constant for air parcels moving around in the troposphere. We call the 10°C/km lapse rate experienced by a parcel moving adiabatically the adiabatic lapse rate.

By contrast, the environmental lapse rate, or the change in temperature vs. altitude in the environment, is dependent on external factors, such as weather patterns and how much the ground is heated by the sun. Its value can be different from day to day or even hour to hour. Typically, the air temperature aloft does not change much during the day, but the temperature of the air at the ground will vary widely from sunrise to sunset because of the heating of the surface air by the sunlight-heated ground. So, in the morning, when the ground temperature is low and there is not much difference between the temperature between the ground and the temperature aloft, the environmental lapse rate is small. In the afternoon, after the sun has heated the ground a lot, there will be a much larger difference between the high ground temperature and the lower air temperature aloft, so the afternoon lapse rate will be larger.