Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 3 - Derivatives - 3.8 Implicit Differentiation - 3.8 Exercises: 25

Answer

\[y = - \frac{5}{4}x + \frac{7}{2}\]

Work Step by Step

\[\begin{gathered} {x^2} + xy + {y^2} = 7\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {2,1} \right) \hfill \\ \hfill \\ use\,\,the\,\,implicit\,\,differentiation \hfill \\ \hfill \\ 2x + xy' + y + 2yy' = 0 \hfill \\ \end{gathered} \] \[factor\,\,y'\] \[y'\,\left( {x + 2y} \right) = - 2x - y\] \[solve\,\,for\,\,y'\] \[y' = - \frac{{2x + y}}{{x + 2y}}\] \[\begin{gathered} evaluate\,\,\,\,\left( {2,1} \right) \hfill \\ \hfill \\ y' = - \frac{{2\,\left( 2 \right) + 1}}{{2 + 2}} = - \frac{5}{4} \hfill \\ \hfill \\ use\,\,the\,\,point\, - \,slope\,\,form \hfill \\ \end{gathered} \] \[\begin{gathered} y - {y_1} = m\,\left( {x - {x_1}} \right) \hfill \\ \hfill \\ then \hfill \\ \hfill \\ y - 1 = - \frac{5}{4}\,\left( {x - 4} \right) \hfill \\ \hfill \\ simplify \hfill \\ \hfill \\ y = - \frac{5}{4}x + \frac{7}{2} \hfill \\ \end{gathered} \]
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.