Answer
\[y = \frac{1}{2}x + 2\]
Work Step by Step
\[\begin{gathered}
\,{\left( {x + y} \right)^{\frac{2}{3}}}\, = y\,\,\,,\,\,\,\,\left( {4,4} \right) \hfill \\
\hfill \\
use\,\,the\,\,Implicit\,\,differentiation \hfill \\
\hfill \\
\frac{2}{3}\,{\left( {x + y} \right)^{ - \frac{1}{3}}}\,\left( {1 + {y^,}} \right) = {y^,} \hfill \\
\hfill \\
distribute \hfill \\
\hfill \\
\frac{2}{{3\sqrt[3]{{x + y}}}} + \frac{{2{y^,}}}{{3\sqrt[3]{{x + y}}}} = {y^,} \hfill \\
\hfill \\
solve\,\,for\,\,{y^,} \hfill \\
\hfill \\
{y^,}\,\left( {\frac{2}{{3\sqrt[3]{{x + y}}}} - 1} \right) = - \frac{2}{{3\sqrt[3]{{x + y}}}} \hfill \\
\hfill \\
{y^,}\, = \frac{{\frac{2}{{3\sqrt[3]{{x + y}}}}}}{{\frac{2}{{3\sqrt[3]{{x + y}}}} - 1}} = - \frac{2}{{2 - 3\sqrt[3]{{x + y}}}} \hfill \\
\hfill \\
find\,\,the\,\,slope \hfill \\
\hfill \\
\,\left( {4,4} \right) \to \,{y^,} = - \frac{2}{{2 - 3\sqrt[3]{{4 + 4}}}} = \frac{1}{2} \hfill \\
\hfill \\
use\,\,the\,\,po\operatorname{int} \,\,slope\,\,form \hfill \\
\hfill \\
y - {y_1} = m\,\left( {x - {x_1}} \right) \hfill \\
\hfill \\
y - 4 = \frac{1}{2}\,\left( {x - 4} \right) \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
y = \frac{1}{2}x + 2 \hfill \\
\end{gathered} \]
