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

and we call the

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

It should now be apparent that

so that

If

then

If then

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].

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

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].

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

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].