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2021-10-22

Use Version 2 of the Chain Rule to calculate the derivatives of the following functions.
$y=\mathrm{sin}\left(2\sqrt{x}\right)$

Asma Vang

Step 1
Consider the function: $y=\mathrm{sin}\left(2\sqrt{x}\right)$
Let .
Version two of chaon rule: $\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$.
Calculate $\frac{dy}{du}$
$\frac{dy}{du}=\frac{d\left(\mathrm{sin}u\right)}{du}$
$=\mathrm{cos}u$
Step 2
Calculate
$\frac{du}{dx}=\frac{d\left(2\sqrt{x}\right)}{dx}$
$=2\cdot \frac{1}{2}{x}^{\frac{1}{2}-1}$
$=\frac{1}{\sqrt{x}}$
Therefore,
$\frac{dy}{dx}=\mathrm{cos}u\cdot \frac{1}{\sqrt{x}}$
$=\frac{\mathrm{cos}\left(2\sqrt{x}\right)}{\sqrt{x}}$

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