How do I find the Jacobi matrix?

Question

I've never done questions like these, so I would very much like some help.

We are given a function $f: \mathbb R^n \to \mathbb R$ given by $f(x)=\langle x,\xi\rangle^2$ where $\langle\,,\rangle$ is the standard inner product of $\mathbb R^n$ and $\xi \in \mathbb R^n$.

Find $D_f(a)$, meaning, the differential of $f$ in point $a$, or in other words, the jacobi matrix multiplied by vector $a$.

Thanks, I would very much like an explanation on how to approach this

Answer 1

Would like someone to review this answer.

let's say $x=\begin{pmatrix} x_1 \\x_2\\ \vdots \\x_n\end{pmatrix}$ and $\xi =\begin{pmatrix} \xi_1 \\\xi_2\\ \vdots \\\xi_n\end{pmatrix}$

then $f(x)=<x,\xi>^2 = (x_1\xi_1+x_2\xi_2+...+x_n\xi_n)^2$

now, instead of actually computing$(x_1\xi_1+x_2\xi_2+...+x_n\xi_n)^2$, we can derive it as such. We can say that the derivative with respect to the variable $x_j$ is: $\frac{d f}{d x_j} = 2\xi_j(x_1\xi_1+x_2\xi_2+...+x_n\xi_n)$