In a Hilbert Space $V$, for function $f:V\to\mathbb{R}$, if $f$ is Fréchet differentiable at $x\_0$, the Fréchet Derivative $\nabla f(x\_0)$ is $v$ such that $$ \lim\_{x\to x\_0} \frac{|f(x) - f(x\_0) - \langle v, x-x\_0 \rangle|}{\|x-x\_0\|} = 0$$

As an example, for $f(x) = \|x\|$, the Fréchet derivative is $\nabla f(x) = \frac{x}{\|x\|}$, as shown in this Wikipedia [proof](https://en.wikipedia.org/wiki/Fr%C3%A9chet_derivative#Example_in_infinite_dimensions).

Now my question is, what is the Fréchet derivative of $f(x) = \frac{1}{\|x\|}$. My gut says something like $-\frac{x}{\|x\|^3}$ following normal derivative rules, but I can't seem to be able to solve it.