I am trying to derive equations where the Taylor expansion about an equilibrium state is involved as following.
$\lvert\lvert \mathbf{x}_{0} + \epsilon \mathbf{x}_{1} \rvert\rvert = \lvert\lvert \mathbf{x}_{0} \rvert\rvert + \epsilon \frac{\mathbf{x}_{1}^{T} \mathbf{x}_{0}}{\lvert\lvert\mathbf{x}_{0}\rvert\rvert}+O\left(\epsilon^2\right)$,
where $\mathbf{x}_{0}$ is an equilibrium solution and $\mathbf{x}_{1}$ is a small perturbation. Is the above expansion correct? Or, is there anything I did wrong?
Thanks,