The following equations are on page 23 in "David Gilbarg, Neil S. Trudinger Elliptic Partial Differential Equations of Second Order, 2nd edition"
$$ \Gamma(x) = \Gamma(|x|) := \begin{cases} \frac{1}{n(2-n)w_n} |x|^{2-n} & n > 2 \\ \frac 1 {2\pi} \log |x| & n=2 \end{cases}$$
$$ D_i \Gamma(x-y) = \frac 1 {nw_n} |x-y|^{-n} (x_i - y_i) $$
My calculation shows $$ D_i \Gamma(x-y) = \frac 1 {nw_n} |x-y|^{1-n} \frac{(x_i - y_i)(x-y)}{|x-y|} $$ This is derived by using $ |u|' = u u' / |u| $. What's wrong with me?
There's a small mistake: $$ D_i\Gamma(x-y)=\frac{1}{n(2-n)w_n}(2-n)|x-y|^{2-n-1}\frac{x_i-y_i}{|x-y|}=\frac{1}{nw_n}|x-y|^{-n}|x-y|\frac{x_i-y_i}{|x-y|}=\frac{1}{nw_n}|x-y|^{-n}(x_i-y_i) $$