Chapter 3 - Derivatives - 3.8 Implicit Differentiation - 3.8 Exercises - Page 200: 49

$y = - 5x + 6$

Work Step by Step

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

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