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The Definition of Higher Order Derivatives

April 2nd, 2009 |

Author: Merrin

This section we will stretch our knowledge of calculus. We have already gained some familiarity with the methods of differentiation and integration. Now we will extend the procedure to repeated differentiations of order n or repeated integrations of order n where n is a positive integer. This is not merely an academic exercise, this will provide us the clues to answer such questions as what is the meaning of

$\cfrac{d^{1/2} }{dx^{1/2} } f(x)$

One also might ask what half order integration might look like. We don’t even have a symbol to write such a monster with at the moment.

Such calculations fall under the name of fractional calculus, for non integer order operations of differentiation and integration. The idea of fractional calculus has been around a long while, and even arose shortly after calculus was developed. In response to a question on what meaning a fractional order derivative might have, Leibnitz wrote in 1665,

$\cfrac{d^{1/2} }{dx^{1/2} }f(x)$

“… an apparent paradox from which some day useful consequences will be drawn.”

Recall our possible notations for derivatives

$\cfrac{d^{n} }{dx^{n} } f(x) \qquad D^{n} f(x) \qquad f^{(n)} (x) \qquad\dddot{f}(x)$

Clearly Newton’s notation for fluxion is not useful because we can’t keep writing dots, or have a fraction of a dot. The most commonly occurring notation is

$D^{n}$

so we will use that, however the Leibnitz notation is also acceptable.

Now as an answer to our first puzzle, we can write a symbol for integration as a derivative of negative order.

$\displaystyle D^{-1} f(x)=\int _{a}^{x}f(x)dx$

Using an operator notation we can make a reasonable calculation that the derivative cancels out the integral when written in a particular order.

$D^{1} D^{-1} f(x)=D^{1-1} f(x)=D^{0} f(x)=f(x)$

Our approach to study fractional calculus will be that taken by others. Derive some general formulas for order n and then generalize them to fractional order replacing factorials with gamma functions and so forth. Then we can check out the properties of the new operations and see if they are consistent with what we expect for certain tests. We can find some closed form expression for the nth order derivatives of many of the elementary functions, in particular those involving exponentials and polynomials. So let us get started.

The definition of the derivative can be extended to higher orders. By now we have left the definition of the derivative in the dust, but the following extension may serve as a starting point for new calculations.

Recall our basic definition of the derivative

$D^{1} f(x)=\mathop{\lim }\limits_{\Delta x\to 0} \cfrac{f(x+\Delta x)-f(x)}{\Delta x}$

The form of the derivative for what follows will be more useful in terms of the back derivative. Approaching from the right or left, what is the difference?

$D^{1} f(x)=\mathop{\lim }\limits_{\Delta x\to 0} \cfrac{f(x)-f(x-\Delta x)}{\Delta x}$

Now let’s develop a formula for the second derivative from the definition.

${D^{2} f(x)}={\mathop{\lim }\limits_{\Delta x\to 0}\cfrac{f(x)-f(x-\Delta x)-(f(x-\Delta x)-f(x-2\Delta x))}{(\Delta x)^{2} } }$
${D^{2} f(x)} ={\mathop{\lim }\limits_{\Delta x\to 0} \cfrac{ f(x)-2f(x-\Delta x)+f(x-2\Delta x) } { (\Delta x)^{2} } }$

What I did was apply the definition for the back difference derivative on each function for the definition of the first derivative. Continuing this way, we can derive the definition for the third derivative and so on.

$D^{3} f(x)=\mathop{\lim }\limits_{\Delta x\to 0} \cfrac{f(x)-3f(x-\Delta x)+3f(x-2\Delta x)-f(x-3\Delta x)}{(\Delta x)^{3} }$

Now what is cool about what we are doing is that we can form a definition for the nth derivative independent of what functions we choose to work on. This rule is evident from above where we see the numbers of the binomial identity cropping up.

$\displaystyle D^{n} f(x)=\mathop{\lim }\limits_{\Delta x\to 0}\cfrac{1}{(\Delta x)^{n} }\sum_{j=0}^{n}(-1)^{j}\cfrac{n!}{j!\left(n-j\right)!}f(x-j\Delta x)$

This relation can be proved by mathematical induction. We already solved the first three cases. This formula could prove useful for numerical analysis where you have to calculate high order derivatives among other things. This equation is also one road to the fractional derivative.

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