For reference this is from Jost's Partial Differential Equations text on pages 13 and 14.
$$\Gamma(x,y):=\Gamma(|x-y|) = \begin{cases}\frac{1}{2\pi}\ln(|x-y|) \; \; if \; d=2 \\ \frac{1}{d(2-d)\omega_d}|x-y|^{2-d} \; \; if \; d>2 \end{cases}$$
$$ G(x,y):= \begin{cases} \Gamma(|x-y|)-\Gamma(\frac{|y|}{R}|x-y|) \; \; \text{for} \;y\not=0 \\ \Gamma(|x|) - \Gamma(R) \; \; \text{for} \; y=0 \end{cases}$$
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 $$ G(x,y)=\Gamma\left((|x|^2+|y|^2-2x\cdot y)^\frac{1}{2} \right) - \Gamma\left( \left(\frac{|x|^2|y|^2}{R^2}+R^2-2x\cdot y \right)^\frac{1}{2}\right) $$
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:
$$ \frac{\partial}{\partial \nu_x}G(x,y)= \frac{\partial}{\partial {|x|}}G(x,y) = \frac{1}{d\omega_d}\frac{|x|}{|x-y|^d}-\frac{1}{d\omega_d}\frac{|x|}{|x-y|^d}\frac{|y|^2}{R^2}$$
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?