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I was looking at fractional calculus on Wikipedia, specifically this section and came across the half derivative of the function $y=x$ which is $y=\frac{2\sqrt{x}}{\sqrt{\pi}}$ . The derivative tells the slope at any point on the curve, but what does the "half derivative" mean - it's obviously not $\frac{1}{2}$ the derivative of $y=x$ which would be just $\frac{1}{2}$.

I do not have a very deep understanding of calculus - I have just taken Calc 1 & 2 out of a 4 series, but anything helps!

Also, I have checked similar questions, but they did not seem to answer my question that I have bolded.

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A related here. Here is geometric explanation –  Mhenni Benghorbal Aug 28 '13 at 4:38

1 Answer 1

up vote 3 down vote accepted

The half derivative itself doesn't have much physical interpretation (though I believe there is a field called fractional quantum mechanics which may use it)

So why does it exist if its not any real physical thing.

I will explain.

Lets consider the idea of counting children at a school. We use whole numbers (positive integers) to count children. The statements are 5,6...201992 children each are meaningful in the sense that they exist mathematically AND have a physical interpretation.

But the set of Numbers isn't just whole numbers. It includes numbers like $1/2$ and $2^{1/2}$. So we could try to ask well what's half a child or square root of 2 children?

These are meaningless questions in the sense that you can't have half a child (contrary to popular belief disassembly and reassembly of children is not an easy or practical thing to do). Irrational quantities are even harder to produce. Put simply, they just DO NOT appear in that context.

So why am I telling you this? Here's why, lets ask the question not what 1/2 means in therms of children but how it came along. It came along because we wanted to generalize the set of numbers to include stuff in between the integers. It came along for applications besides counting children and is in fact most specifically an "accidental-byproduct" of the existence of division.

So what's a fractional derivative? We can easily answer the question that the nth derivative is the "rate of change of the rate of change ... (Repeat n times) of the rate of change of the function". This like children is a discrete structure. Only whole numbers (and if you include integrals then negative integers as well (like a backwards derivative)) work.

The fractional derivative is a consequence of the question "what is the function whom I apply twice to get a first derivative". Rather than "what is the rate of... Rate of change of the function"

So in short. It's an interesting question where we extend our level of control and understanding of calculus but it shares little similarity with the more physical forms that calculus originally had.

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I see someone down voted this ( or two people actually since it's -2) I would be more than willing to answer any questions if you'd like to bring them up –  frogeyedpeas Jun 16 at 15:40

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