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Answered question

2021-10-22

Use Version 2 of the Chain Rule to calculate the derivatives of the following functions.

y = \sin (2 \sqrt{x})

Answer & Explanation

Asma Vang

Skilled2021-10-23Added 93 answers

Step 1
Consider the function:

y = \sin (2 \sqrt{x})

Let

u = 2 \sqrt{x} t h e n y (u) = \sin u

.
Version two of chaon rule:

\frac{d y}{d x} = \frac{d y}{d u} \frac{d u}{d x}

.
Calculate

\frac{d y}{d u}

\frac{d y}{d u} = \frac{d (\sin u)}{d u}

= \cos u

Step 2
Calculate

\frac{d u}{d x} b y a p p l y \in g \frac{d (a x^{n})}{d x} = a n x^{n - 1}

\frac{d u}{d x} = \frac{d (2 \sqrt{x})}{d x}

= 2 \cdot \frac{1}{2} x^{\frac{1}{2} - 1}

= \frac{1}{\sqrt{x}}

Therefore,

\frac{d y}{d x} = \cos u \cdot \frac{1}{\sqrt{x}}

= \frac{\cos (2 \sqrt{x})}{\sqrt{x}}

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