Answer
\[y = - x + 2\]
Work Step by Step
\[\begin{gathered}
{x^3} + {y^3} = 2xy\,\,\,\,\,\,,\,\,\,\left( {1,1} \right) \hfill \\
\hfill \\
use\,\,the\,\,implicit\,\,differentiation \hfill \\
\hfill \\
3{x^2} + 3{y^2}{y^,} = 2x{y^,} + 2y \hfill \\
\hfill \\
collect\,\,like\,\,terms \hfill \\
\hfill \\
3{y^2}{y^,} - 2x{y^,} = 2y - 3{x^2} \hfill \\
\hfill \\
solve\,\,for\,\,y{\,^,} \hfill \\
\hfill \\
{y^,}\,\left( {3{y^2} - 2x} \right) = 2y - 3{x^2} \hfill \\
\hfill \\
{y^,} = \frac{{2\,\left( 1 \right) - 3\,{{\left( 1 \right)}^2}}}{{3\,{{\left( 1 \right)}^2} - 2\,\left( 1 \right)}} = 1 \hfill \\
\hfill \\
use\,\,the\,\,point\, - \,slope\,\,form \hfill \\
\hfill \\
y - {y_1} = m\,\left( {x - {x_1}} \right) \hfill \\
\hfill \\
then \hfill \\
\hfill \\
y - 1 = - 1\,\left( {x - 1} \right) \hfill \\
\hfill \\
y - 1 = - x + 1 \hfill \\
\hfill \\
y = - x + 2 \hfill \\
\hfill \\
\end{gathered} \]
