Let $\mathcal{A}$ be a Banach Algebra. Let $I \subset \mathbb{R}$ be an open interval. Let $g,f: I\to \mathcal{A}$ be differentiable function. Then I know that $fg$ is again differentiable with
\begin{equation} (fg)'=f'g+fg'. \end{equation} My first question would be if Leibniz's formula for the nth derivative does also apply in this case?
My second question is whether there is a formula for \begin{equation} \frac{d^k}{d\lambda^k}f^n, \end{equation} where $k,n\in \mathbb{N}$ and $f: I\to \mathcal{A}$ is $k$ times differentiable. I found a formula for the real case here:
https://www.physicsforums.com/threads/nth-derivative-of-f-x-n.418767/.
But I can not find a reference.