Answer
\[y = - \frac{2}{9}x + \frac{{17}}{9}\]
Work Step by Step
\[\begin{gathered}
x{y^{\frac{5}{2}}} + {x^{\frac{3}{2}}}y = 12\,\,\,\,,\,\,\,\left( {4,1} \right) \hfill \\
\hfill \\
use\,\,the\,\,Implicit\,\,differentiation \hfill \\
\hfill \\
x\,\left( {\frac{5}{2}} \right){y^{\frac{3}{2}}}{y^,} + {y^{\frac{5}{2}}} + {x^{\frac{3}{2}}}{y^,} + \frac{3}{2}{x^{\frac{1}{2}}}y = 0 \hfill \\
\hfill \\
Collect\,\,like\,\,terms \hfill \\
\hfill \\
{y^,}\,\left( {\frac{5}{2}x{y^{\frac{3}{2}}} + {x^{\frac{3}{2}}}} \right) = - {y^{\frac{5}{2}}} - \frac{3}{2}{x^{\frac{1}{2}}}y \hfill \\
\hfill \\
therefore \hfill \\
\hfill \\
{y^,} = \frac{{ - {y^{\frac{5}{2}}} - \frac{3}{2}{x^{\frac{1}{2}}}y}}{{\frac{5}{2}x{y^{\frac{3}{2}}} + {x^{\frac{3}{2}}}}} \hfill \\
\hfill \\
evaluate\,\,\left( {4,1} \right) \hfill \\
\hfill \\
{y^,} = \frac{{\, - {{\left( 1 \right)}^{\frac{1}{2}}} - \frac{3}{2}\,{{\left( 4 \right)}^{\frac{1}{2}}}\,\left( 1 \right)}}{{\frac{5}{2}\,\left( 4 \right)\,{{\left( 1 \right)}^{\frac{3}{2}}} + \,{{\left( 4 \right)}^{\frac{3}{2}}}}} \hfill \\
\hfill \\
use\,\,the\,\,point\,\,slope\,\,form \hfill \\
\hfill \\
y - {y_1} = m\,\left( {x - {x_1}} \right) \hfill \\
\hfill \\
y - 1 = - \frac{2}{9}\,\left( {x - 4} \right) \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
y - 1 = - \frac{2}{9}x + \frac{8}{9} \hfill \\
\hfill \\
y = - \frac{2}{9}x + \frac{{17}}{9} \hfill \\
\end{gathered} \]
