I am having the craziest thought about calculus:

So we know that $\dfrac{d}{dx}x^n = nx^{n-1}$ and $\dfrac{d^2}{dx^2}x^n = n(n-1)x^{n-2}$

So... What about $\dfrac{d^{1/2}}{dx^{1/2}}x^n$? Is it even possible?

Question

I am having the craziest thought about calculus:

So we know that $\dfrac{d}{dx}x^n = nx^{n-1}$ and $\dfrac{d^2}{dx^2}x^n = n(n-1)x^{n-2}$

So... What about $\dfrac{d^{1/2}}{dx^{1/2}}x^n$? Is it even possible?

dan815 · Answer

haha good question

anonymous · Answer

@satellite73 @iambatman

anonymous · Answer

I believe degree should be $n-\frac{1}{2}$, but that's just wild guess... 

What about coefficient?

Guess nobody will know...

anonymous · Answer

@e.mccormick @freckles Any idea on how to start such thing?

anonymous · Answer

http://en.wikipedia.org/wiki/Fractional_calculus

anonymous · Answer

This in particular: http://en.wikipedia.org/wiki/Fractional_calculus#Fractional_derivative_of_a_basic_power_function

anonymous · Answer

Looks scary... I mean I saw $_aD_t^\alpha f(t)=\frac{d^n}{dt^n} {}_aD_t^{-(n-\alpha)}f(t)=\frac{d^n}{dt^n} {}_aI_t^{n-\alpha} f(t)$

Guess I am kind of not ready :(

anonymous · Answer

but dang I don't know it is already considered. lol

anonymous · Answer

Ye lol, I was asking myself: how would you define half a derivative? It has to be something close to half a power that you can use power rules to extend from integer powers.
They really discuss this kind of thinking in the Heuristics section =)