Originally posted on MSE.

Let $U$ be an open set in $\mathbb{C}^{n}$ let $F$ be a Banach space (in my case even a dual Banach space), and let $\varphi:U\to F$ be a holomorphic map. I seem to be able to prove that the differential map $D\varphi:U\times\mathbb{C}^{n}\to F$ defined by $$D\varphi (z,v)= \lim\limits_{t\to 0}\frac{\varphi(z+tv)-\varphi(z)}{t}$$ is holomorphic.

Is there a reference for this assertion? (Or at least for continuity)

I tried to look into some sources on infinite-dimensional holomorphicity and could not find such a statement, but some of those sources are rather complicated, and so it is likely I missed it.