Fractional derivatives (FractionalD) of an InterpolatingFunction to retun another InterpolatingFunction

Question

Question

I want to calculate the FractionalD of an InterpolatingFunction object, however, evaluating

FractionalD[Interpolation[Range[5]][x],{x,1/2}]

evaluates the argument but FractionalD remains unchanged, returning

FractionalD[ InterpolatingFunction[(* Something *)] ]

I would have expected an InterpolatingFunction object, the same output as Interpolation[Sqrt[Range[5]]*2/Sqrt[Pi]]

How to calculate the fractional derivative of an InterpolatingFunction so the output is another InterpolatingFunction?

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Background

Fractional Derivatives

On the one hand, Wolfram Language (Mathematica) is fully capable of performing symbolic calculations of Fractional Derivatives defined as

For example,using FractionalD, FractionalD[f[x],{x,α}] gives the Riemann–Liouville fractional derivative of order α of the function $f(x)$,

FractionalD[x,{x,1/2}]

Interpolating Functions

On the other hand, one can take derivatives of InterpolatingFunction objects using D and Derivative.

For example,

D[Interpolation[Range[9]][x],x]

Visually,

Plot[
    Evaluate[
        {
            Interpolation[Range[5]][x],
            D[Interpolation[Range[5]][x],x]
        }
    ]
    ,{x,0, 3}
]

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Due diligence

Web

I have searched the site and the web for the relevant keywords, unsuccessfully.

Piecewise (edited)

Thanks to @MichaelE2way for pointing out that @CarlWoll 's InterpolationToPiecewise transforms an InterpolatingFunction (only "Hermite" Method) into a Piecewise object. In that particular case, the answer would be more straightforward, as FractionalD does evaluate over Piecewise functions and one could re-build the InterpolatingFunction with FunctionInterpolation. I'm still interested in alternative solutions.

NFractionalD

It seems NFractionalD does work with InterpolatingFunction

NFractionalD[Interpolation[Range[9]][x], {x, 1/2}, 0.25]

But that would imply that I need to deconstruct the InterpolatingFunction to extract the data points to use NFractionalD in each point to then build a new InterpolatingFunction using Interpolation. That looks too cumbersome and error-prone.