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Chapter 3: Problem 9

Use Version I of the Chain Rule to calculate \(\frac{d y}{d x}\). $$y=\sin ^{5} x$$

Short Answer

Expert verified
Question: Find the derivative of the function \(y = \sin^5 x\) with respect to \(x\). Answer: The derivative of the function \(y = \sin^5 x\) with respect to \(x\) is \(\frac{dy}{dx} = 5(\sin x)^4 \cdot \cos x\).

Step by step solution

01

Define the inner and outer functions

We write the given function as \(y = (\sin x)^5\) and define: - The inner function \(f(x) = \sin x\) - The outer function \(g(x) = x^5\)
02

Compute the derivatives of the inner and outer functions

We need to find the derivatives of the inner and outer functions: - \(f'(x) = \frac{d(\sin x)}{dx} = \cos x\) - \(g'(x) = \frac{d(x^5)}{dx} = 5x^4\)
03

Apply the Chain Rule

Now we can use the Chain Rule to find the derivative of the composition \(y = g(f(x))\). According to the Chain Rule: $$\frac{dy}{dx} = g'(f(x)) \cdot f'(x)$$ So, substituting the expressions for \(g'(x)\) and \(f'(x)\) and \(f(x)\), we get: $$\frac{dy}{dx} = 5(\sin x)^4 \cdot \cos x$$

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