Let \((P_\vartheta)_{\vartheta \in \Theta}\) be a parametric family of Markov kernels from a measurable space \((X, \mathcal{X})\) to a locally compact space \(Y\). The family \((P_\vartheta)_{\vartheta \in \Theta}\) is called weakly differentiable at \(\vartheta\) if for any \(x \in X\) there is a finite signed Baire measure \(P'_\vartheta(x, .)\) on \(Y\) such that \[ \frac{d}{d\vartheta} \int g(y)P_\vartheta(x, dy)=\int g(y)P'_\vartheta(x, dy) \] for all continuous test functions \(g\) with compact support on \(Y\). In the paper, the authors give sufficient conditions for such a weak derivative \(P'_\vartheta\) to possess a representation as a scaled difference of two Markov kernels. Moreover, this Jordan type decomposition is explicitly constructed.