If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by
$$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$
Let $f$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$. If $f$ is of class $\mathcal{C}^1$, show that $(T_f)' = T_{f'}$. Adapt this result to the case where $f$ is piecewise of class $\mathcal{C}^1$.