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College
Calculus and Analysis
Differential Calculus
Derivatives

Rolando Wade

Answered question

2022-03-22

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

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

Answer & Explanation

kattylouxlvc

Beginner2022-03-23Added 11 answers

Step 1
Consider the function: $y = \sin (2 \sqrt{x})$
Let $u = 2 \sqrt{x} t hen y (u) = \sin u$ .
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 $\frac{d u}{d x} by applying \frac{d (a x^{n})}{d x} = a n x^{n - 1}$
$\frac{d u}{d x} = \frac{d (2 \sqrt{x})}{d x}$
$= 2 \cdot \frac{1}{2} x^{\frac{1}{2} - 1}$
$= \frac{1}{\sqrt{x}}$
Therefore,
$\frac{d y}{d x} = \cos u \cdot \frac{1}{\sqrt{x}}$
$= \frac{\cos (2 \sqrt{x})}{\sqrt{x}}$

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