Fractional Differential

Question

How would you take a fractional differential. Such that $f^n(x)=3x^2+5x-4$ where $n=1/2$. I have been told that this could be done; however, when I try and figure it out I am not coming up with a reasonable answer.

Answer 1

Because $\ f^{(n)}(x)= D^{n}f(x)$, where $\ D=d/dx$,

and $$\ D^{n}f(x) = \frac{1}{\Gamma(1-n)}\frac{d}{dx} \int_0^x \frac{f(t)}{(x-t)^n} dt, 0<n<1$$

hence $$ f^{(0.5)}(x)= \frac{1}{\Gamma(0.5)}\frac{d}{dx} \int_0^x \frac{3t^2+5t−4}{\sqrt{x-t}} dt$$ $$ \Rightarrow f^{(0.5)}(x)= \frac{1}{\sqrt{\pi}}\frac{d}{dx} \begin{bmatrix} \frac{4}{15}(12x^{2.5}+25x^{1.5}-30x^{0.5})\end{bmatrix}$$ $$ \Rightarrow f^{(0.5)}(x)= \frac{8x^{1.5}+10x^{0.5}-4x^{-0.5}}{\sqrt{\pi}}$$