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# 15.1 Higher Order Derivatives

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