Let $ L:\mathbb{R}^n\to\Bbb{R}$ be the function $L(x)=\langle x|y\rangle$ , where $\langle|\rangle$ denotes some inner product on $\Bbb{R}^n$ and y is a fixed vector in $\Bbb{R}^n$. Further denote by DL, the derivative of $L$. Then which of the following are necessarily correct?

1. $ DL(u)= DL(v)$ for all $u, v \in \Bbb R^n$.
2. $DL(0,0,...,0)= L$.
3. $DL(1,1,...,1)= 0$.
4. $DL(x)=||x||^2,\forall x\in\Bbb R^n$.

By the definition of linear functionals. It seems $L$ is a linear functional.
$$L(x)=<x|y>=<(x\_1,...,x\_n)|(y\_1,...,y\_n)>=\sum\_{i=1}^n x\_iy\_i$$

How to use $D$ in this problem?

I saw some answers in this app. But every answer choose correct options as incorrect options.

Correct answers are (1) and (2).