Calculus: Early Transcendentals (2nd Edition)

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

Chapter 3 - Derivatives - 3.7 The Chain Rule - 3.7 Exercises: 10

Answer

\[ = - 5\cdot {x^4}\sin {x^5} \]

Work Step by Step

\[\begin{gathered} y = \cos {x^5} \hfill \\ \hfill \\ y = f\,\left( u \right) = \cos u \hfill \\ \hfill \\ set\,\,u = g\,\left( x \right) = {x^5} \hfill \\ \hfill \\ Use\,\,the\,\,version\,\,1\,\,of\,\,the\,\,chain\,\,rule \hfill \\ \hfill \\ {\text{ }}\frac{{dy}}{{dx}} = \frac{{dy}}{{du}} \cdot \frac{{du}}{{dx}} \hfill \\ \hfill \\ Therefore \hfill \\ \hfill \\ \frac{{dy}}{{dx}} = \frac{d}{{du}}\,\left( {\cos u} \right) \cdot \frac{d}{{dx}}\,\left( {{x^5}} \right) \hfill \\ \hfill \\ = - \sin u \cdot 5{x^4} \hfill \\ \hfill \\ substitute\,\,\,back\,\,u \hfill \\ \hfill \\ = - 5\cdot {x^4}\sin {x^5} \hfill \\ \end{gathered} \]
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