Use the Chain Rule to find the derivative. \( y=\cos ^{3}(12 x) \) \( y^{\prime}=\square \)

Let \( y = \cos^3(12x) \). 1. Let \( u = \cos(12x) \). Then, \( y = u^3 \). 2. Compute the derivative of \( y \) with respect to \( u \): \[ \frac{dy}{du} = 3u^2 \] 3. Compute the derivative of \( u \) with respect to \( x \). Since \( u = \cos(12x) \), using the chain rule: \[ \frac{du}{dx} = -\sin(12x) \cdot 12 = -12\sin(12x) \] 4. Apply the chain rule: \[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 3u^2 \cdot (-12\sin(12x)) \] 5. Substitute back \( u = \cos(12x) \): \[ y' = -36\cos^2(12x)\sin(12x) \] Thus, the derivative is: \[ y' = -36\cos^2(12x)\sin(12x) \]