Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 2 - Derivatives - 2.4 Derivatives of Trigonometric Functions - 2.4 Exercises - Page 150: 28

Answer

$$\frac{\sin \left(x\right)}{2\sqrt{x}}+\sqrt{x}\cos \left(x\right)$$

Work Step by Step

Given $$ f(x)=\sqrt{x}\:\sin \:x $$ Since \begin{aligned} f'(x)&=\:\frac{d}{dx}\left(\sqrt{x}\:\sin \:x\:\:\right)\\ &=\frac{d}{dx}\left(\sqrt{x}\right)\sin \left(x\right)+\frac{d}{dx}\left(\sin \left(x\right)\right)\sqrt{x}\\ &=\frac{\sin \left(x\right)}{2\sqrt{x}}+\sqrt{x}\cos \left(x\right) \end{aligned} We can note that $f(x) $ is decreasing when $f'(x)<0$ and $f(x) $ is increasing when $f'(x)>0 $
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