let $D_{\mathbb N}$ be the standard "n-th derivative" function

is it possible to make a continuation of $D_{\mathbb N}$ to non integer values?

i mean a function $D_{\mathbb R}$ such that $D_{\mathbb R}(x,f)=D_{\mathbb N}(n,f)$ for all $x=n\in\mathbb N$

it should be something relevant, linear interpolation usually doesn't make any sense.