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: 3

Answer

\[\frac{{dy}}{{dx}} = \frac{{5y - 8x}}{{3y - 5x}}\]

Work Step by Step

\[\begin{gathered} 8{x^2} - 10xy + 3{y^2} = 26 \hfill \\ \,Find\,\,dy/dx\,\,\,by\,\,implicit\,\,differentiation \hfill \\ \frac{d}{{dx}}\,\left( {8{x^2} - 10xy + 3{y^2}} \right) = \frac{d}{{dx}}\,\left( {26} \right) \hfill \\ Use\,\,the\,\,product\,\,rule\,\,for\,\,\frac{d}{{dx}}\,\left( {10xy} \right) \hfill \\ 16x - 10\,\left( {x{y^,} + y\,\left( 1 \right)} \right) + 6y{y^,} = 0 \hfill \\ Multiplying \hfill \\ 16x - 10x{y^,} - 10y + 6y{y^,} = 0 \hfill \\ Move\,\,all\,\,{y^,}\,\,to\,\,the\,\,same\,\,side\,\,of\,\,the\,\,equation \hfill \\ - 10x{y^,} + 6y{y^,} = 10y - 16x \hfill \\ Factor \hfill \\ {y^,}\,\left( { - 10x + 6y} \right) = 10y - 16x \hfill \\ {y^,} = \frac{{10y - 16x}}{{6y - 10x}} \hfill \\ Then \hfill \\ \frac{{dy}}{{dx}} = \frac{{5y - 8x}}{{3y - 5x}} \hfill \\ \end{gathered} \]
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