The fractional derivative of course returns the usual derivative when the derivative "powers" add up to an integerpic.twitter.com/GlPN2iKf9k
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Fractional Calculus
The 𝑛ᵗʰ derivative can be generalized to the 𝑞ᵗʰ derivative, where 𝑞 is a rational number
For polynomials 𝑥ⁿ, this is done by replacing the factorial function with its continuous generalization: the Γ-functionpic.twitter.com/KAGh6nMcrk
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This extension allows us to ask what happens when we take the "negative oneth" derivative
We would like to obtain the "left-cancellation" property of the derivative
To do this we are forced into the definition that the "negative oneth" derivative being the antiderivativepic.twitter.com/EPVXapmrpo
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Why does q have to be rational? It doesn't look there is a rationality restriction from the formula here, but I just don't know.
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Good observation! Ιt doesn't have to be rational! I just didn't want to jump in to the complexes right off the bat
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what’s half the derivative of exp(x) ?
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You can either (1) expand in a power series and use the above formula, or (2) use the more general definition of fractional derivative
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Is this used somewhere in physics or is it just a cool generalization that can be done?
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Indeed it's used in physics! See
@dblanch256's retweet about the Brachistochrone problem - 1 more reply
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Also notice it is not always "distributive". The 5/2-derivative of x isn't 0! It's interesting to look at when this is actually true. What if you split 5/2 into 5/4+5/4. Will it be true then? 7/4+3/4? Etcpic.twitter.com/daCakGpOhl
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Looks like the 5/4-derivative of the 5/4-derivative of x = 7/4-derivative of the 3/4-derivative of x = 5/2-derivative of x. Maybe you just can't separate across an integer?
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