Find the distributional limit of the sequence of distributions
$$F_n=n\delta_{-\frac{1}{n}}-n\delta_{\frac{1}{n}}$$
Hint: $F_n$ is the distributional derivative of some function
So far I've tried kind of working backwards towards the definition of the distributional derivative to make use of the hint:
$$F_n=n\delta_{-\frac{1}{n}}-n\delta_{\frac{1}{n}}$$
$$F_n=<n\delta_{-\frac{1}{n}},\phi>-<n\delta_{\frac{1}{n}},\phi>$$
$$n\int_{-1/n}^{1/n}\phi'(x) \ dx$$
$$-\int_{\mathbb{R}}f(x)\phi'(x) \ dx$$
$$-<f,\phi'> \ = \ <f',\phi>$$
where
$$f(x) =
\begin{cases}
-n & \text{$-\frac{1}{n}\le x\le\frac{1}{n}$} \\
0 & \text{otherwise}
\end{cases}$$
But I'm not sure how to proceed from here and how following this hint got me closer to the answer.
Any help is greatly appreciated!
Take some test function $\phi$, then
$$\langle F_n, \phi\rangle = \int n\big(\delta(x+\tfrac 1n) - \delta(x-\tfrac 1n)\big)\phi(x) d x = n\big(\phi(-\tfrac 1n)-\phi(+\tfrac 1n)\big) \overset{n\to\infty}{\longrightarrow} -2\phi'(0) $$
So the limit distribution $F$ is the linear functional $F(\phi) = -2\phi'(0)$. In particular $F=2\delta'$.