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: 40

Answer

\[\frac{{dy}}{{dx}} = {e^x}\,\left( {\frac{3}{2}\sqrt x + \sqrt {{x^3}} } \right)\]

Work Step by Step

\[\begin{gathered} y = {e^x}\sqrt {{x^3}} \hfill \\ \hfill \\ differentiate \hfill \\ \hfill \\ \frac{{dy}}{{dx}} = \,\left( {{e^x}} \right)\,\left( {\frac{{3{x^2}}}{{2\sqrt {{x^3}} }}} \right) + \sqrt {{x^3}} {e^x} \hfill \\ \hfill \\ therefore \hfill \\ \hfill \\ \frac{{dy}}{{dx}} = {e^x}\,\left( {\frac{3}{2}{x^{\frac{1}{2}}}} \right) + \sqrt {{x^3}} {e^x} \hfill \\ \hfill \\ factor \hfill \\ \hfill \\ \frac{{dy}}{{dx}} = {e^x}\,\left( {\frac{3}{2}\sqrt x + \sqrt {{x^3}} } \right) \hfill \\ \hfill \\ \end{gathered} \]
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