Answer
$R_s(1,2)=32, R_t(1,2)=-39$
Work Step by Step
We will use the Chain Rule to find $R_s(s,t)$ and $R_t(s,t)$ and then we will find it for $s=1$ and $t=2$:
$$R_s(s,t)=\frac{\partial}{\partial s}(G(u(s,t),v(s,t)))=G_u(u(s,t),v(s,t))u_s(s,t)+G_v(u(s,t),v(s,t))v_s(s,t)$$
$$R_t(s,t)=\frac{\partial}{\partial t}(G(u(s,t),v(s,t)))=G_u(u(s,t),v(s,t))u_t(s,t)+G_v(u(s,t),v(s,t))v_t(s,t)$$
So we have:
$$R_s(1,2)=G_u(u(1,2),v(1,2))u_s(1,2)+G_v(u(1,2),v(1,2))v_s(1,2)=
G_u(5,7)\cdot4+G_v(5,7)\cdot2=9\cdot4-2\cdot2=32$$
$$R_t(1,2)=G_u(u(1,2),v(1,2))u_t(1,2)+G_v(u(1,2),v(1,2))v_t(1,2)=
G_u(5,7)\cdot(-3)+G_v(5,7)\cdot6=9\cdot(-3)-2\cdot6=-39$$
