Question
Find the derivative of the trigonometric function. \[ y=4 \sin (6 \pi x) \] \( y^{\prime}(x)=\square \)
Ask by Collins Jimenez. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The derivative is \( y'(x) = 24\pi \cos(6\pi x) \).
Solution
To differentiate y = 4 sin(6πx), we'll use the chain rule. The derivative of sin(u) with respect to x is cos(u) · u′, where u = 6πx. Since u′ = 6π, we have:
y′(x) = 4 · cos(6πx) · (6π) = 24π cos(6πx).
Thus, the derivative is:
y′(x) = 24π cos(6πx).
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Simplify this solution The Deep Dive
To find the derivative of the function \( y = 4 \sin(6 \pi x) \), we use the chain rule. The outer function is \( 4 \sin(u) \) where \( u = 6 \pi x \). The derivative of \( \sin(u) \) is \( \cos(u) \), and we also need to account for the derivative of \( u \) which is \( 6 \pi \). Thus, we have: \[ y' = 4 \cos(6 \pi x) \cdot (6 \pi) = 24 \pi \cos(6 \pi x) \] So, \( y^{\prime}(x) = 24 \pi \cos(6 \pi x) \).
