What Is the Half Derivative of (ax+c)^(3/2)?

[Jocher]

d^1/2(ax+c)^(3/2)/dx^(1/2)=? please help me :)

 

[Simon Bridge]

It is tempting to start by taking everything under the common square root, isn't it?
However, you have probably figured out that this does not work for the derivative operator.

Just like when y=x^{1/2}, you are actually looking for y: y^2=x, when you see D^{1/2}f you are actually looking to use something like Hf(x) <img src="/styles/physicsforums/xenforo/smilies/arghh.png" class="smilie" loading="lazy" alt=":H" title="Gah! :H" data-shortname=":H" />^2f(x)=Df(x).

Look for http://en.wikipedia.org/wiki/Fractional_calculus]fractional[/PLAIN] calculus[/itex].

 

[JJacquelin]

d^1/2(ax+c)^(3/2)/dx^(1/2)=? please help me :)

In using the Riemann-Liouville operator for the definition of the fractionnal derivative, and in order to avoid somme difficulties of integral convergence, it will be simpler to :
First, compute the fractional integral (order 1/2) of the function.
Then, compute the classical derivative.

 

[JJacquelin]

Even simpler, by performing first the classical derivation, then the fractionnal integration (joint page)
(be carefull with this manner, which is generally not recommended. But in the present case, it works)

 

[JJacquelin]

You're welcome !

I used fractionnal derivatives in complex impedance analysis.
The particular case of half derivative is common in modeling the electrical properties of many homogeneous compounds (a particular behaviour and impedance, long ago loosely refreed as Warburg's impedance).
Especially, the fractionnal derivatives of the sinusoidal functions are interesting.
For example, in the paper "The Phasance Concept", pp.5-6
http://www.scribd.com/JJacquelin/documents