illusiia

2021-10-22

Find the derivatives of the functions $y=\sqrt{2x}\mathrm{sin}\left(\sqrt{x}\right)$ .

krolaniaN

Skilled2021-10-23Added 86 answers

Step 1

Given function is$y=\sqrt{2x}\mathrm{sin}\left(\sqrt{x}\right)$ .

Step 2

Compute the derivative.

${y}^{\prime}=\sqrt{2}\left[\sqrt{x}\mathrm{sin}\sqrt{x}\right]$

$=\sqrt{2}[\sqrt{x}{\left(\mathrm{sin}\sqrt{x}\right)}^{\prime}+{\left(\sqrt{x}\right)}^{\prime}\mathrm{sin}\sqrt{x}]$

$=\sqrt{2}[\sqrt{x}\left(\mathrm{cos}\sqrt{x}\right)\left(\frac{1}{2}{x}^{-\frac{1}{2}}\right)+\left(\frac{1}{2}{x}^{-\frac{1}{2}}\right)\mathrm{sin}\sqrt{x}]$

$=\sqrt{2}[\frac{\sqrt{x}\mathrm{cos}\sqrt{x}}{2\sqrt{x}}+\frac{\mathrm{sin}\sqrt{x}}{2\sqrt{x}}]$

$=\frac{\sqrt{2}}{2}[\mathrm{cos}\sqrt{x}+\frac{\sqrt{x}\mathrm{sin}\sqrt{x}}{x}]$

Given function is

Step 2

Compute the derivative.

Jeffrey Jordon

Expert2022-08-30Added 2605 answers

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