Suppose I have a function $f(x) = A^n$ where $A$ is a square matrix, $x$ is a positive real scalar, and $n$ is a natural number.
I would like to calculate the derivative of $f$ with respect to $x$ (each entry in $A$ is a function of $x$).
Is there a simple formula for this in general or do I need to know what $n$ is and use the product rule?
I found this, but I don't understand it (in particular I don't understand what $DS(A)$ or $S(A)$ means).
edit: Each entry in $A$ is differentiable.