Let $L$ be a finite extension of $\mathbb Q_p$ with ring of integers $\mathcal{O}=\mathcal{O}_L$ and let $B_1(L):=\{z \in L \colon \vert z-1 \vert <1 \}$. Let $\widehat{\mathcal{O}}(L)_{\mathbb Q_p}$ denote the set of all locally $\mathbb Q_p$-analytic characters on $\mathcal{O}$ with values in $L$. It can be shown, that \begin{align} B_1(L) \otimes_{\mathbb Z_p} \text{Hom}_{\mathbb Z_p}(\mathcal{O}, \mathbb Z_p) & \to \widehat{\mathcal{O}}(L)_{\mathbb Q_p} \\ z \otimes \beta & \mapsto [l \mapsto z^{\beta(l)}] \end{align} is a bijection. Denote the character associated to $z \otimes \beta$ under this map by $\chi_{z,\beta}$. We also define the map \begin{align} d\colon \widehat{\mathcal{O}}(L)_{\mathbb Q_p} & \to \text{Hom}_{\mathbb Q_p}(L,L) \\ \chi & \mapsto d\chi, \end{align} where $d\chi(a):=\frac{d}{dt}\chi(t \cdot a)\vert_{t=0}$ for $a \in L$.
I want to show that
\begin{align} d\chi_{z,\beta}=log(z)\cdot \beta. \end{align}
In my calculations I keep coming up with $d\chi_{z,\beta}(a)=log(z)\cdot \beta'(0)\cdot a$, which is not right. I would appreciate someone showing me the right way to calculate this.