Q1: What is the correct definition of the derivative of a distribution $T \in \mathcal{D}^\prime(\mathbb{R})$? Here we write $T^\prime$ for the distributional derivative.

A1: $T^{\prime}(\varphi) = - T(\varphi^{\prime}) $

A2: $T^{\prime}(\varphi) = T(\varphi) $

A3: $T^{\prime}(\varphi) = \varphi^{\prime}$

A4: $T^{\prime}(\varphi) = T(\varphi^{\prime}) $

A5: $T^{\prime}(\varphi) = - T(\varphi) $

Q2: What is the correct definition of the partial derivative of a distribution $T \in \mathcal{D}^\prime(\mathbb{R}^n)$?

A1: $D^\alpha T(\varphi) = (-1)^{|\alpha|} T( D^\alpha \varphi) $

A2: $D^\alpha T(\varphi) = T( D^\alpha \varphi) $

A3: $D^\alpha T(\varphi) = - T( D^\alpha \varphi) $

A4: $D^\alpha T(\varphi) = (-1)^{|\alpha|} T( \varphi) $

A5: $D^\alpha T(\varphi) = (-1)^{|\alpha|+1} T( D^\alpha \varphi) $

Q3: Consider the regular distribution given by $$ f(x) = \begin{cases} -1 \text{ if } x \leq 0 \ \hphantom{-}1 \text{ if } x > 0 \end{cases} $$ For $\alpha = (1)$, what is the distributional derivative of $T_f$?

A1: $D^\alpha T_f = 2 \delta $

A2: $D^\alpha T_f = - \delta $

A3: $D^\alpha T_f = \delta $

A4: $D^\alpha T_f = 3 \delta $