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: 60

Answer

\[ = {\sec ^2}\,\left( {x{e^x}} \right) \cdot \,\left( {{e^x} + x{e^x}} \right)\]

Work Step by Step

\[\begin{gathered} y = \,\tan \,\left( {x{e^x}} \right) \hfill \\ \hfill \\ differentiate\, \hfill \\ \hfill \\ {y^,} = \frac{d}{{dx}}\,\left( {\tan \,\left( {x{e^x}} \right)} \right) \hfill \\ \hfill \\ use\,\,the\,\,\,Chain\,\,rule \hfill \\ \hfill \\ = {\sec ^2}\,\left( {x{e^x}} \right) \cdot \,{\left( {x{e^x}} \right)^,} \hfill \\ \hfill \\ use\,\,the\,Product\,\,rule \hfill \\ \hfill \\ = {\sec ^2}\,\left( {x{e^x}} \right) \cdot \,\left( {{x^,} \cdot {e^x} + x\, \cdot {{\left( {{e^x}} \right)}^,}} \right) \hfill \\ \hfill \\ simplify \hfill \\ \hfill \\ = {\sec ^2}\,\left( {x{e^x}} \right) \cdot \,\left( {{e^x} + x{e^x}} \right) \hfill \\ \end{gathered} \]
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