$A$ is an $m\times n$ matrix and $b$ is an $m \times 1$ column vector. Vector-valued $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is given by $f(x) = Ax + b$. Find the derivative, $f'(x)$.
I was able to solve for the derivative of $f: \mathbb{R} \rightarrow \mathbb{R}^m$, $f(x) = ax^2 + b$, where $a$ and $b$ are column vectors in $\mathbb{R}^m$. Not sure how to start this problem. This is for homework; hints would be appreciated.
If $$f(x)=\begin{bmatrix}a_{11}x_1+\cdots +a_{1n}x_n\\ a_{21}x_1+\cdots +a_{2n}x_n\\ \vdots \\ a_{n1}x_1+\cdots +a_{nn}x_n\end{bmatrix}+\mathbf{b}$$ the gradient of the first output is $$\langle a_{11},a_{12},...,a_{1n} \rangle$$ and similarly for the other gradients. So what then would be the derivative?
As far as the interpretation, note that the function is linear, and that derivatives provide best linear approximations. In this light the result should make perfect sense.