Answer
\[y = - 5x + 6\]
Work Step by Step
\[\begin{gathered}
xy + {x^{\frac{3}{2}}}{y^{ - \frac{1}{2}}} = 2\,\,\,\,\,,\,\,\,\,\left( {1,1} \right) \hfill \\
\hfill \\
use\,\,the\,\,Implicit\,\,differentiation \hfill \\
\hfill \\
x{y^,} + y + {x^{\frac{3}{2}}}\,\left( { - \frac{1}{2}} \right){y^{ - \frac{3}{2}}}{y^,} + \frac{3}{2}{x^{\frac{1}{2}}}{y^{ - \frac{1}{2}}} = 0 \hfill \\
\hfill \\
Collect\,\,the\,\,like\,\,terms \hfill \\
\hfill \\
{y^,}\,\left( {x - \frac{1}{2}{x^{\frac{3}{2}}}{y^{ - \frac{3}{2}}}} \right) = - y - \frac{3}{2}{x^{\frac{1}{2}}}{y^{ - \frac{1}{2}}} \hfill \\
\hfill \\
therefore \hfill \\
\hfill \\
{y^,} = \frac{{ - y - \frac{3}{2}{x^{\frac{1}{2}}}{y^{ - \frac{1}{2}}}}}{{x - \frac{1}{2}{x^{\frac{3}{2}}}{y^{ - \frac{3}{2}}}}} \hfill \\
\hfill \\
\,find\,\,the\,\,slope\,\,use\,\,\left( {1,1} \right) \hfill \\
\hfill \\
{y^,} = \frac{{ - 1 - \frac{3}{2}}}{{1 - \frac{1}{2}}} = - 5 \hfill \\
\hfill \\
use\,\,the\,\,point\,\,slope\,\,form \hfill \\
y - {y_1} = m\,\left( {x - {x_1}} \right) \hfill \\
\hfill \\
therefore \hfill \\
\hfill \\
y - 1 = - 5\,\left( {x - 1} \right) \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
y - 1 = - 5x + 5 \hfill \\
\hfill \\
y = - 5x + 6 \hfill \\
\hfill \\
\end{gathered} \]
