Consider a matrix $n\times n$ $A$ which depends on a $n\times n$ matrix $B$ i.e., $A=A(B)$. The matrix $B$ in turn depends upon a continuous real parameter $\phi$ such that for small $\phi$, $B$ can be written as $$B(\delta\phi)=\mathbb{1}-C\delta\phi$$ where $C$ is also a $n\times n$ matrix.
Can we expand $A(B(\delta\phi))=A(\mathbb{1}-C\delta\phi)$ in a Taylor series?
I would simplify your question in this way: how a matrix with entries $a_{ij}=a_{ij}(t)$ depending on a real parameter $t$ can be expanded into a Taylor series ?
There is a simple answer : expand separately all the $a_{ij}(t)$s into Taylor series (around the origin or around another point, if they can be expanded...), then factorize $1,t,t^2...$.
The best is to see it on an example :
$$\begin{pmatrix}\cos(t)&\sin(t)\\0&1\end{pmatrix}=I_2+t\begin{pmatrix}0&1\\0&0\end{pmatrix}+t^2/2\begin{pmatrix}-1&0\\ \ \ 0&0\end{pmatrix}+\cdots$$