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.

Hint: The entries of $X(t)$ are obviously differentiable, and the entries of $f(t)$ are polynomials in the entries of $X(t)$. 
