Let $D$ be an open set in $\Bbb{R}^n$ and $$u :D\to L(\Bbb{R}^n,\Bbb{R}^n)$$ be a function of class $C^k$ for $k\geq 1.$ Let $$f :D\to \Bbb{R}^n$$ be defined by $f(x)=u(x)(x).$
I want to show that $f$ is of class $C^{k}$ and compute $f'$
Definition: A function $f : U \to V$ is said to be of class $C^k$ if the first $k$ derivatives $f'(x),f''(x),\cdots,f^k(x)$ all exist and are continuous.
I'm still uncertain on how to go about this. Please, can anyone render help or provide a reference? Thanks in advance!
Let $u(x)$ be represented by an $n\times n$ matrix with components $A_{ij}(x)$. Since $u$ is $C^k$, it follows that each $A_{ij}: D \rightarrow \mathbb{R}^n$ is $C^k$. For $x=(x_1,\ldots, x_n) \in \mathbb{R}^n$ and $f(x)=(f_1(x),\ldots, f_n(x))$, we have $$f_i(x)=\sum_{j=1}^n A_{ij}(x)x_j,$$ which is clearly $C^k$ because it is a sum of $C^k$ functions. To find the derivative, use sum and product rules: $$\dfrac{\partial f_i}{\partial x_k}(x)=A_{ik}+\sum_{j=1}^n \dfrac{\partial A_{ij}(x)}{\partial x_k}x_j$$