Show that $f:I\to\mathbb{R}^{n^2}$ defined by $f(t)=X(t)^k$ is differentiable Let $I$ be a interval, $\mathbb{R}^{n^2}$ be the set of all $n\times n$ matrices and $X:I \to\mathbb{R}^{n^2}$ be a differentiable function. Given $k\in\mathbb{N}$, define $f:I\to\mathbb{R}^{n^2}$ by $f(t)=X(t)^k$. How to prove that $f$ is differentiable?
Thanks.

$X$ is differentiable and $f$ is a matrix product of $X$, $k$ number of times. So $f$ is also differentiable.
