This is part b to a question about a function, $f:\mathbf{R}^{n}\rightarrow\mathbf{R}^{n}$, where $f(x) = \frac{x}{|x|}$. The first question is to find the Jacobian, $\partial f(x)$, which I found to have entries $a_{ij}=\frac{\delta_{ij}}{|x|}-\frac{x_ix_j}{|x|^{3}}$, where $\delta_{ij}$ denotes the entry in the corresponding identity matrix.
Consider $\Pi_{x}:=\{v\in\mathbf{R}^{n}:x\cdot v = 0\}$ (ie, the set of vectors orthogonal to x). The question is to prove that $\Pi_{x}$ is an eigenspace of $\partial f(x)$.
I know that, for $\Pi_{x}$ to be an eigenspace, for all $v \in \Pi_x$, $(\partial f(x) - \lambda I_n)v = 0$, for some $\lambda$ which is an eigenvalue of $\partial f(x)$.
Beyond that, I am unsure of how to approach the question.
Edited to correct notation
Let us prove that the hyperplane $\Pi_{x}$ is an eigenspace of $\partial f(x)$ (not of $\partial f$) with eigenvalue $\frac1{\|x\|},$ starting from the Jacobian $A=\partial f(x)$ (not $\partial f$) which you computed.
If $x\cdot v=0,$ i.e. $\sum_jx_jv_j=0,$ then the $i$-th component of $Av$ is $$(Av)_i=\sum_ja_{i,j}v_j=\frac{v_i}{\|x\|}-\frac{x_i}{\|x\|^2}\sum_jx_jv_j=\frac{v_i}{\|x\|}. $$