Vtyjkyuk

For reference this is from Jost's Partial Differential Equations text on pages 13 and 14.

$$Gamma(x,y):=Gamma(|x-y|) = begincasesfrac12piln(|x-y|) ; ; if ; d=2
\ frac1d(2-d)omega_d|x-y|^2-d ; ; if ; d>2
endcases$$

$$ G(x,y):= begincases Gamma(|x-y|)-Gamma(fracR|x-y|) ; ; textfor ;ynot=0 \
Gamma(|x|) - Gamma(R) ; ; textfor ; y=0
endcases$$

Above, $Gamma$ is the fundamental solution of the Laplace equation in $mathbb R^d$, $omega_d$ is the volume of the unit ball in $mathbb R^d$, and $G$ is the Green's function for the Dirichlet problem on a ball of radius $R$ centered at $0$, ie. $B_R(0)$

We may rewrite $G$ as such
$$ G(x,y)=Gammaleft((|x|^2+|y|^2-2xcdot y)^frac12 right) - Gammaleft( left(fracyR^2+R^2-2xcdot y right)^frac12right) $$

Jost then says that for $x in partial B_R(0)$, since we can rewrite $G$ as we just did above than then we can calculate the directional derivitive with respect to the exterior normal $nu$ as follows:

$$ fracpartialpartial nu_xG(x,y)= fracpartialpartial xG(x,y) = frac1domega_dfracx^d-frac1domega_dfracx^dfracyR^2$$

My first question is what is a text book that explains how to take a directional derivative like the one above. I feel like I am lacking some knowledge here. Would Spivak's Calculus on Manifolds be of help?

And if possible can someone explain how this derivative is calculated?