The derivative refers to the rate of change, or slope, of a function.
Differentiation refers to the calculation of a derivative.

Derivatives are used throughout applied mathematics and science. The first
derivative most students come across is speed, which is the rate of change
of position with respect to time (60 miles per hour means that if you
were to keep driving at the same speed for one hour your position
would change by 60 miles).

The derivative of a function is the instantaneous rate of
change of a function evaluated at each point. This is written
as

How are Derivatives Calculated?

Analytic Calculation

While it is possible to verify many formulas for calculating
derivatives using the above definition, most of the time derivatives
are calculated using a look-up table. Since a table including
every single derivative in existance would be a bit cumbersome, often
only the most fundamental derivatives are listed, and one is required
to apply a variety of useful rules to calculate more complicated derivatives.

Some common derivatives are:

Some of the more common rules regarding derivative calculation are:

k = constant

Numerical Calculation

Another method for calculating derivatives involves approximating the exact
limit above with a difference:
, h > 0.

More complicated methods involve assuming a known simple function, such as a
polynomial, can be fit to data at a few points.