Es ist
\(\displaystyle|x| = \sqrt{\sum_{i=1}^n x_i^2} = \left(\sum_{i=1}^n x_i^2\right)^{\frac{1}{2}}\)
und somit
\(\displaystyle\frac{\partial |x|}{\partial x_i} = 2x_i\cdot\frac{1}{2}\left(\sum_{i=1}^n x_i^2\right)^{-\frac{1}{2}}=\frac{x_i}{|x|}\),
also
\(\displaystyle\frac{\partial^2 |x|}{\partial x_i\partial x_j} = x_i\cdot 2x_j\cdot \left(-\frac{1}{2}\right)\cdot\left(\sum_{i=1}^n x_i^2\right)^{-\frac{3}{2}}=-\frac{x_ix_j}{|x|^3}\)
für \(i\neq j\) und
\(\begin{aligned}\frac{\partial^2 |x|}{\partial x_i^2} &= \left(\sum_{i=1}^n x_i^2\right)^{-\frac{1}{2}} + x_i\cdot 2x_i\cdot\left(-\frac{1}{2}\right)\cdot \left(\sum_{i=1}^n x_i^2\right)^{-\frac{3}{2}}\\&=\frac{1}{|x|}-\frac{x_i^2}{|x|^3}\text{.}\end{aligned}\)
