As usual in such examples there is no need to integrate against a test function. It is a simple consequence of the fact that if a sequence (or net) of distributions converges in the distributional sense, then the sequence obtained by differentation also converges. In particular, this applies if the original sequence consists of functions which converge in pretty well any sensible classical sense, e.g., locally $L^1$ as is the case here.
In this example we consider the functions which are defined as $x^{\epsilon}$ on the positive real axis, and $0$ elsewhere. They converge to the Heaviside function in the above sense and so we can differentiate to obtain the required result. Most of the representations of $\delta$-sequences can be obtained in this way: consider the primitives and show they form a sequence which converges to the Heaviside function in a suitable sense. The result then follows as above.