Let \( f(x)=4 \sin ^{2} 3 x \) find the value of \( f^{\prime}\left(\frac{1}{4} \pi\right) \)

To find the value of \( f^{\prime}\left(\frac{1}{4} \pi\right) \) for the function \( f(x) = 4 \sin^2(3x) \), follow these steps: 1. **Differentiate the Function:** Start by differentiating \( f(x) \) with respect to \( x \): \[ f(x) = 4 \sin^2(3x) \] Use the chain rule for differentiation. Recall that: \[ \frac{d}{dx} \sin^2(u) = 2 \sin(u) \cos(u) \cdot \frac{du}{dx} \] Applying this to \( f(x) \): \[ f'(x) = 4 \cdot 2 \sin(3x) \cos(3x) \cdot 3 = 24 \sin(3x) \cos(3x) \] Alternatively, using the double-angle identity \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \): \[ f'(x) = 12 \sin(6x) \] 2. **Evaluate at \( x = \frac{1}{4} \pi \):** Using the simplified derivative: \[ f'\left(\frac{1}{4} \pi\right) = 12 \sin\left(6 \cdot \frac{1}{4} \pi\right) = 12 \sin\left(\frac{3}{2} \pi\right) \] Calculate \( \sin\left(\frac{3}{2} \pi\right) \): \[ \sin\left(\frac{3}{2} \pi\right) = \sin(270^\circ) = -1 \] Therefore: \[ f'\left(\frac{1}{4} \pi\right) = 12 \times (-1) = -12 \] **Final Answer:** \( f^{\prime}\left(\frac{1}{4} \pi\right) = -12 \)