Periodic distribution of a piecewise continous function

Question

Let $f$ be a periodic piecewise continous function with period $2\pi$, and its derivative (classical derivative) is also a piecewise continous function and we denote by $\frac{df}{dx}$. Let $x_{1},..,x_{n}$ discontinuity points of $f$ in $(-\pi,\pi]$.Then its distributional derivative $f'$ is given by $$f'=\frac{df}{dx}+\sum\limits_{j=1}^{n}[f(x_{j}^{+})-f(x_{j}^{-})]\delta_{x_{j}}.$$ $f(x_{j}^{+})=\lim_{x \downarrow x_{j}} f(x),f(x_{j}^{-})=\lim_{x \uparrow x_{j}} f(x) $ and $\delta_{x_{j}} $ is the dirac function centered at $x_{j}\in \mathbb{R}$
Note that $f$ defines a periodic distribution by the formula $$\langle f,\varphi\rangle=\int_{-\pi}^{\pi}f(x)\varphi(x)dx$$ Well, Can you help me understand what $\frac{df}{dx}$ means in the distributional derivative? and How to pove that the distributional derivative has that form?