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5. Functions of the Derivative

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It is worth to mention that being (4.4) the expression of the a-st derivative of a function the derivative itself is

 (5.1)

And being the a-st derivative or iterating the differentiation a times powering it to a, applying other functions to the derivative could be also considered. If the function applied to the derivative can be expanded in powers of x,

 (5.2)

The result is a weighted sum of different order derivatives. These functions of the derivative are usually known as "formal differential operators". As an example, the exponential of the derivative applied to the exponential would give the following result that could be valuable for calculating functions of the derivative when both f and g can be expanded in exponentials

 (5.3)

If both functions f and g can be expanded in positive powers of x,

 (5.4)

Thus,

 (5.5)

At a first glance this seems not very interesting, but interesting properties could be hidden under this apparent mess. We shall return later to this point with the help of other tools.