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Chapter 3: Problem 16

Use Version I of the Chain Rule to calculate \(\frac{d y}{d x}\). $$y=\sin \frac{x}{4}$$

Short Answer

Expert verified
Answer: The derivative of the function \(y = \sin\frac{x}{4}\) with respect to \(x\) is \(\frac{dy}{dx} = \frac{1}{4}\cos\frac{x}{4}\).

Step by step solution

01

Identify the inner function

In the given function \(y = \sin\frac{x}{4}\), the inner function is \(v(x) = \frac{x}{4}\). We will denote \(u(v) = \sin v\) as the outer function.
02

Find the derivative of the outer function with respect to the inner function

We need to find \(\frac{dy}{dv}\). Since \(y = u(v) = \sin v\), this is simply the derivative of \(u(v)\) with respect to \(v\): $$\frac{dy}{dv} = \frac{d(\sin v)}{dv} = \cos v$$
03

Find the derivative of the inner function with respect to x

We need to find \(\frac{dv}{dx}\). Since \(v(x) = \frac{x}{4}\), this is simply the derivative of \(v(x)\) with respect to \(x\): $$\frac{dv}{dx} = \frac{d(\frac{x}{4})}{dx} = \frac14$$
04

Apply the Chain Rule

Now we can apply the Chain Rule by multiplying the derivatives from Steps 2 and 3: $$\frac{dy}{dx} = \frac{dy}{dv}\cdot\frac{dv}{dx} = \cos v \cdot \frac{1}{4}$$
05

Replace v with the inner function

Finally, we substitute the inner function \(v(x) = \frac{x}{4}\) back into the expression for the derivative: $$\frac{dy}{dx} = \cos\left(\frac{x}{4}\right)\cdot\frac{1}{4}$$ Therefore, the derivative of \(y = \sin\frac{x}{4}\) with respect to \(x\) is: $$\frac{dy}{dx} = \frac{1}{4}\cos\frac{x}{4}$$

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