Since the derivative of a function
*f*
at any value
*x*
_{0}
where the function is
differentiable is just a number,
*f'*(*x*
_{0})
, we may define the derivative function,
denoted
*f'*(*x*)
or
(*x*)
, to be the function that maps a value
*x*
where
*f*
is differentiable to the derivative of
*f*
at
*x*
. Since
*f'*
is a function in
its own right, we can graph it, just like any other function.

For example, let us calculate the derivative function (also called simply the
"derivative") of
*f* (*x*) = 5*x*
. Since the graph of this function is a line with slope
5
, we might guess that the derivative will equal
5
at every point. We verify this
with the formula for the
derivative of
*f*
at a point
*x*
:

f'(x) |
= | ||

= | |||

= | 5 |

Indeed,
*f'*(*x*) = 5
, a constant function.

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