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 \begin{aligned} \partial_{x} R &=\frac{p}{\alpha^{2}} \frac{\partial_{x} G}{1+G}+\frac{1}{\alpha} D(x), \\ \partial_{y} R &=\frac{p}{\alpha^{2}} \frac{\partial_{y} G}{1+G}-\frac{1}{\alpha} D(y), \\ \partial_{x} \partial_{y} R &=\frac{p}{\alpha^{2}} \frac{\partial_{x} \partial_{y} G(1+G)-\left(\partial_{x} G\right)\left(\partial_{y} G\right)}{(1+G)^{2}}, \\ \partial_{x}^{2} R &=\frac{p}{\alpha^{2}} \frac{\partial_{x}^{2} G(1+G)-\left(\partial_{x} G\right)^{2}}{(1+G)^{2}}+\frac{1}{\alpha} D^{\prime}(x), \\ \partial_{y}^{2} R &=\frac{p}{\alpha^{2}} \frac{\partial_{y}^{2} G(1+G)-\left(\partial_{x} G\right)^{2}}{(1+G)^{2}}-\frac{1}{\alpha} D^{\prime}(y), \\ \partial_{x} G &=-\alpha\left\{G+\frac{D(x)}{p}\right\}, \\ \partial_{y} G &=\frac{\alpha}{p} e^{\alpha(y-x)} D(y), \\ \partial_{x} \partial_{y} G &=-\alpha \partial_{y} G, \\ \partial_{x}^{2} G &=-\alpha\left\{-\alpha G-\alpha \frac{D(x)}{p}+\frac{D^{\prime}(x)}{p}\right\}, \end{aligned}