Can we use Cauchy's repeated integral formula to differentiate functions?
$$(I^nf)(t)=\frac{1}{\Gamma(n)}\int_a^x (x-t)^{n-1}\space f(t)\space\space dt,\space\space n\notin \mathbb Z_{\leqslant 0}$$
If we replace $n$ with $-n$ such that $0<n<1$, do we end up finding $n$-th derivative of $f(t)$?
I tried to evaluate this equation by trying to find half derivative of $x$, but the integral diverged so I didn't success. Can we use this to differentiate functions and how is it applied on $f(t)=t$