There is a theorem in fractional derivative that states that the Caputo Derivative of order $\alpha >0$ with $n-1<\alpha<n$ of the power function $f(t) = t^p$ for $p≥0$ satisfies:
$$\operatorname{D}_t^\alpha t^p = \frac{\Gamma{(p+1)}}{\Gamma{(p-\alpha+1)}}$$ when $(p>n-1)$ , and $$\operatorname{D}_t^\alpha t^p = 0$$ when $ (p≤n-1)$.
My question is how do you find the Caputo Derivative of $\operatorname{D}_t^\alpha t^{-\alpha}$ when $0<\alpha<1$ since the theorem above does not apply in this case. Thanks for your response in advance.