Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 6 - Applications of the Derivative - 6.4 Implicit Differentiation - 6.4 Exercises: 8

Answer

\[\frac{{dy}}{{dx}} = \frac{5}{{2y\,\left( {5 - {x^2}} \right)}}\]

Work Step by Step

\[\begin{gathered} 2{y^2} = \frac{{5 + x}}{{5 - x}} \hfill \\ \,Find\,\,dy/dx\,\,\,by\,\,implicit\,\,differentiation \hfill \\ \frac{d}{{dx}}\,\left( {2{y^2}} \right) = \frac{d}{{dx}}\,\left( {\frac{{5 + x}}{{5 - x}}} \right) \hfill \\ Use\,\,the\,\,power\,\,rule\,\,and\,\,quotient\,\,rule \hfill \\ 4y{y^,} = \frac{{\,\left( {5 - x} \right)\,\left( 1 \right) - \,\left( {5 + x} \right)\,\left( { - 1} \right)}}{{\,{{\left( {5 - x} \right)}^2}}} \hfill \\ 4y{y^,} = \frac{{5 - x + 5 + x}}{{\,{{\left( {5 - x} \right)}^2}}} \hfill \\ 4y{y^,} = \frac{{10}}{{\,{{\left( {5 - x} \right)}^2}}} \hfill \\ {y^,} = \frac{{10}}{{4y\,{{\left( {5 - x} \right)}^2}}} \hfill \\ Then \hfill \\ \frac{{dy}}{{dx}} = \frac{5}{{2y\,\left( {5 - {x^2}} \right)}} \hfill \\ \hfill \\ \end{gathered} \]
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