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15.1
Definition (Higher order derivatives.)
Let
be a function whose domain is a subset of
R. We define a function
(called the
derivative of ) by
and for all
, the value of
at
is the
derivative
. We may also write
for
. Since
is itself a function, we can calculate its derivative: this
derivative is denoted by
or
, and is called
the
second derivative of . For integers
we define
|
(15.2) |
and we call
the
th derivative of . We also
define
In Leibniz's notation
we write
so that equation (
15.2) becomes
If and are real numbers, and and are functions
then from known properties of the derivative we can show that
or
15.3
Examples.
If
, where
, then
It should now be apparent that
so that
If
then
If then
15.4
Exercise.
Calculate
if
.
15.5
Exercise.
A
Let
. Calculate
,
,
and
in terms
of
,
,
,
and
.
What do you think is the formula for
?
15.6
Exercise.
Find
if
.
15.7
Exercise.
Find
if
.
15.8
Exercise.
A
Suppose
for all
.
What can you say about
?
15.9
Exercise.
A
Let
and
be functions such that
and
are defined on
all of
. Show that
Find a similar function for
(assuming that
and
are defined.)
In Leibniz's calculus, or was actually
an infinitely
small quantity that was so much smaller than that the quotient
was zero, and
was obtained
by multiplying by itself and then dividing the result into .
Leibniz also used notations like
and
for which our modern notation has no counterparts.
Leibniz considered the problem of defining a meaning
for
, but he did not make much progress
on this problem. Today there is considerable literature
on
fractional derivatives. A brief history of the subject
can be found in [36, ch I and ch VIII].
15.10
Exercise.
Let
be a real number.
Show that for
|
(15.11) |
After doing this it should be clear that
equation (
15.11), in fact holds for all
(this
can be proved by induction). Now suppose that
and we will
define
|
(15.12) |
Show that then for all
and
in
,
Find
and
.
What do you think
should be?
Equation (15.12) was the starting point from which Joseph Liouville (1809-1882) developed a theory of fractional calculus[36, pp 4-6].
15.13
Exercise.
A
Let
and
be real numbers. Show that for
|
(15.14) |
After doing this exercise it should be clear that in fact
equation (
15.14) holds for all
(this can
be proved by induction).
Now suppose that
, and we will
define
|
(15.15) |
Show that for all
and
in
Equation (
15.15) was used as the starting point for a definition
of fractional derivatives for general functions,
by Joseph Fourier (1768-1830)[
36, page 3].
Next: 15.2 Acceleration
Up: 15. The Second Derivative
Previous: 15. The Second Derivative
  Index
Ray Mayer
2007-09-07