If I have the following function: $a\textbf{x}$ , where $a$ and $\textbf{x}$ are both vectors that I am taking the dot product between, how can I differentiate this function with respect to $\textbf{x}$? What about the 2nd derivative?
For a concrete example, let's assume a = [1, 2, 3] and $\textbf{x} \in \mathbb{R}^3$ (I think this is the correct terminology to say that $\textbf{x}$ can be any vector of length 3 that is made up of real numbers?).
I'm asking because on Math Stack Exchange I see many complicated examples, and I just need some intuition about the very basics.
The derivative of $a\cdot \textbf{x}$ with respect to $\textbf{x}$ is $$ \left({\partial\over dx},{\partial\over dy}\right)a\cdot \textbf{x}=(a_x,a_y). $$ It is not so very different than the derivative of $f(x)=cx$, ${d\over dx}f(x)=c$.