If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by
$$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$
We consider the map $T$ which associates to every function $\varphi$ of $\mathcal{D}$ the real number $T(\varphi)$ defined by
$$T(\varphi) = \int_{-1}^{0} t\varphi(t) \mathrm{d}t + \int_{0}^{+\infty} \varphi(t) \mathrm{d}t$$
a) Show that $T$ is a regular distribution.\\
b) Calculate the derivative of this distribution.