Answer
$u_{x_1}=\frac{x_1}{\sqrt{x_1^2+x_2^2+...x_n^2}}$, $u_{x_2}=\frac{x_2}{\sqrt{x_1^2+x_2^2+...x_n^2}}$,,, $u_{x_n}=\frac{x_n}{\sqrt{x_1^2+x_2^2+...x_n^2}}$.
Work Step by Step
$u=\sqrt{x_1^2+x_2^2+...x_n^2}$
In order to find $u_{x_1}$ we treat all other variables as constants and differentiate with respect to $x_1$.
$u_{x_1}=\frac{x_1}{\sqrt{x_1^2+x_2^2+...x_n^2}}$
Analogously:
$u_{x_2}=\frac{x_2}{\sqrt{x_1^2+x_2^2+...x_n^2}}$
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$u_{x_n}=\frac{x_n}{\sqrt{x_1^2+x_2^2+...x_n^2}}$
