Convergence of the derivative of a BV function in sense of measures

Suppose I have a smooth sequence $f_{n} : \mathbb{R} \to \mathbb{R}$ with $f_{n} \to f$ strongly in $L^{1}_{loc}(\mathbb{R})$ and $f \in BV_{loc}(\mathbb{R})$. Is there any way to justify that $\partial_{x}f_{n} \rightharpoonup \partial_{x}f$ in the sense of measures, i.e. that $$ \int_{K}\phi ~d(\partial_{x}f_{n}) \to \int_{K}\phi ~d(\partial_{x}f)$$ for any compact $K$?

If not then I wonder, given $f \in BV_{loc}(\mathbb{R})$ does there exist a sequence $f_{n}$ which converges in some sense to $f$ and also $\partial_{x}f_{n} \rightharpoonup \partial_{x}f$ in the sense of measures?