Let $f:U\longrightarrow \mathbf{C}$ be a holomorphic function, where $U$ is a Riemann surface, e.g., $U=\mathbf{C}$, $U=B(0,1)$ or $U$ is the complex upper half plane, etc.
For $a$ in $\mathbf{C}$, let $t_a:\mathbf{C} \longrightarrow \mathbf{C}$ be the translation by $a$, i.e., $t_a(z) = z-a$.
What is the difference between $df$ and $d(t_a\circ f)$ as differential forms on $U$?
My feeling is that $df = d(t_a\circ f)$, but why?