A preliminary remark. The operator $(d/dx)^\gamma$ is never continuous on the Schwartz space (and thus on tempered distributions) except if $\gamma$ is a non-negative integer, since you introduce a singularity at 0 by multiplication by $(i\omega)^\gamma=\exp(\gamma \log(i\omega))$ (you have to choose a determination of the logarithm on the imaginary axis and you get a singularity).
Now you can also define what is called an homogeneous distribution with degree $\lambda$, where $\lambda$ is a given complex number: a distribution $T$ on $\mathbb R$ is said to be homogeneous with degree $\lambda$ whenever $$ x\frac{d T}{dx}=\lambda T. $$ It is an exercise to prove that an homogeneous distribution is actually tempered. Examples are $$ \chi_{+,\lambda}=(x_+)^{\lambda}/\Gamma(\lambda +1),\quad \chi_{-,\lambda}=(x_-)^{\lambda}/\Gamma(\lambda +1), $$ and it is possible to prove that homogenous distributions of degree $\lambda$$\lambda\notin \mathbb Z_-$ are $$ c_{+}\chi_{+,\lambda}+c_{-}\chi_{-,\lambda} \quad\text{where $c_\pm$ are constants.} $$$$ c_{+}\chi_{+,\lambda}+c_{-}\chi_{-,\lambda} \quad\text{where $c_\pm$ are constants.} \tag{$\ast$} $$ Note that for $\lambda=-1$, homogeneous distributions of degree $-1$ on the real line are linear combinations of $ \text{pv}\frac{1}{x},\ \delta_0. $ With $\mathscr S'_\lambda$ standing for homogeneous distributions with degree $\lambda$, we get that for $\lambda, \lambda-\gamma\notin \mathbb Z_-$, $$ D^\gamma:\mathscr S'_\lambda\longrightarrow \mathscr S'_{\lambda-\gamma}. $$ To prove this you check that $ (d/dx)^\gamma\chi_{+,\lambda}=\chi_{+,\lambda-\gamma}. $
N.B. Multi-dimensional versions are available, more information in Lars Hörmander's ALPDO 256, Section 3.2.