Solution
= Solution
The <distributional derivative> is defined by moving the <derivative> onto a <test function> and changing the sign:
$$
\boxed{\langle S',\phi\rangle=-\langle S,\phi'\rangle.}
$$
Since $\langle\delta_a,\phi\rangle=\phi(a)$, one has $\langle\delta_0',\phi\rangle=-\phi'(0)$. Hence
$$
\boxed{T=-\delta_0'+\delta_1=-\delta'(x)+\delta(x-1).}
$$
The minus sign in front of $\delta_0'$ is essential: it makes its action the positive <derivative> evaluation required here.