Answer
\[y = \frac{2}{3}x + \frac{5}{3}\]
Work Step by Step
\[\begin{gathered}
{x^4} - {x^2}y + {y^4} = 1\,\,\,\,\,\,\,\,,\,\,\,\left( { - 1,1} \right) \hfill \\
\hfill \\
use\,\,the\,\,implicit\,\,differentiation \hfill \\
\hfill \\
4{x^3} - {x^2}y' - 2xy + 4{y^3}y' = 0 \hfill \\
\hfill \\
factor\,\,{y^,} \hfill \\
\hfill \\
y'\,\left( {4{y^3} - {x^2}} \right) = 2xy - 4{x^3} \hfill \\
\hfill \\
solve\,\,for\,\,y{\,^,} \hfill \\
\hfill \\
y' = \frac{{2xy - 4{x^3}}}{{4{y^3} - {x^2}}} \hfill \\
\hfill \\
evaluate\,\,the\,\,point\,\,\left( { - 1,1} \right) \hfill \\
\hfill \\
y' = \frac{{2\,\left( { - 1} \right)\,\left( 1 \right) - 4\,{{\left( { - 1} \right)}^3}}}{{4\,{{\left( 1 \right)}^3} - \,{{\left( { - 1} \right)}^2}}} \hfill \\
\hfill \\
use\,\,the\,\,point\, - \,slope\,\,form \hfill \\
\hfill \\
y - {y_1} = m\,\left( {x - {x_1}} \right) \hfill \\
\hfill \\
then \hfill \\
\hfill \\
y - 1 = \frac{{2\,\left( {x + 1} \right)}}{3} \hfill \\
\hfill \\
y = \frac{2}{3}x + \frac{5}{3} \hfill \\
\hfill \\
\end{gathered} \]
