Given a function
*f*
, we have defined its derivative
*f'*
. This allows us to now define
*f''*
, the *second derivative* of
*f*
, to be the derivative of
*f'*
. In terms of the
graph of
*f*
, the second derivative tells how quickly the slope of the graph is changing
near a particular point. Alternately, the second derivative tells how quickly the rate
of change of a function is changing.

Since
*f''*
is again a function, we can define
*f'''* = *f*
^{(3)}
to be the derivative of
*f''*
. Continuing in this way, we can define the
*nth*
derivative of
*f*
,
*f*
^{(n)}
.
The derivative of a differentiable function need not be differentiable everywhere
(or even anywhere), so in general the domains of the successive derivatives
*f*
^{(}
*n*)
shrink as
*n*
increases.