Knight Vega
01/29/2024 · High School
3.5. Find the derivatives as indicated for the following : a) \( y=\sin ^{3}(4 x) \quad ; \frac{d y}{d x} \)
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Tutor-Verified Answer
Step-by-step Solution
Find the first order derivative with respect to \( x \) for \( \sin^{3}(4x) \).
Evaluate the derivative by following steps:
- step0: Evaluate the derivative:
\(\frac{d}{dx}\left(\sin^{3}\left(4x\right)\right)\)
- step1: Use differentiation rules:
\(\frac{d}{dg}\left(g^{3}\right)\times \frac{d}{dx}\left(\sin\left(4x\right)\right)\)
- step2: Find the derivative:
\(3g^{2}\times \frac{d}{dx}\left(\sin\left(4x\right)\right)\)
- step3: Calculate:
\(3g^{2}\times 4\cos\left(4x\right)\)
- step4: Substitute back:
\(3\sin^{2}\left(4x\right)\times 4\cos\left(4x\right)\)
- step5: Multiply the terms:
\(12\sin^{2}\left(4x\right)\cos\left(4x\right)\)
The derivative of \( y = \sin^{3}(4x) \) with respect to \( x \) is \( \frac{dy}{dx} = 12\sin^{2}(4x)\cos(4x) \).
Quick Answer
\( \frac{d y}{d x} = 12\sin^{2}(4x)\cos(4x) \)
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