Answer
${{\bf{T}}_u}{|_{\left( {2,3} \right)}} = \left( {4,1,1} \right)$
${{\bf{T}}_v}{|_{\left( {2,3} \right)}} = \left( { - 6,1, - 1} \right)$
${\bf{N}}\left( {2,3} \right) = - 2{\bf{i}} - 2{\bf{j}} + 10{\bf{k}}$
The equation of the tangent plane:
$x + y - 5z = 5$
Work Step by Step
We have $G\left( {u,v} \right) = \left( {{u^2} - {v^2},u + v,u - v} \right)$. So,
${{\bf{T}}_u} = \frac{{\partial G}}{{\partial u}} = \left( {2u,1,1} \right)$, ${\ \ \ }$ ${{\bf{T}}_u}{|_{\left( {2,3} \right)}} = \left( {4,1,1} \right)$
${{\bf{T}}_v} = \frac{{\partial G}}{{\partial v}} = \left( { - 2v,1, - 1} \right)$, ${\ \ \ }$ ${{\bf{T}}_v}{|_{\left( {2,3} \right)}} = \left( { - 6,1, - 1} \right)$
${\bf{N}}\left( {2,3} \right) = {{\bf{T}}_u} \times {{\bf{T}}_v} = \left| {\begin{array}{*{20}{c}}
{\bf{i}}&{\bf{j}}&{\bf{k}}\\
4&1&1\\
{ - 6}&1&{ - 1}
\end{array}} \right| = - 2{\bf{i}} - 2{\bf{j}} + 10{\bf{k}}$
The equation of the tangent plane to the surface at the point $G\left( {2,3} \right) = \left( { - 5,5, - 1} \right)$ is
$\left( {x + 5,y - 5,z + 1} \right)\cdot\left( { - 2, - 2,10} \right) = 0$
$ - 2\left( {x + 5} \right) - 2\left( {y - 5} \right) + 10\left( {z + 1} \right) = 0$
$x + 5 + y - 5 - 5z - 5 = 0$
The equation can be written as
$x + y - 5z = 5$