1. The problem gives $x=\cos 4\theta$ and $y=b\sin 4\theta$ and asks to find $\frac{dy}{dx}$ and examine the options. 2. Differentiate $x$ and $y$ with respect to $\theta$: $$\frac{dx}{d\theta} = -4\sin 4\theta$$ $$\frac{dy}{d\theta} = 4b\cos 4\theta$$ 3. Use the chain rule: $$\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{4b\cos 4\theta}{-4\sin 4\theta} = -b \frac{\cos 4\theta}{\sin 4\theta} = -b \cot 4\theta$$ 4. Now, since $\cot 4\theta = \frac{1}{\tan 4\theta}$, $$\frac{dy}{dx} = -b \cot 4\theta = -\frac{b}{\tan 4\theta}$$ 5. The options mention $-b\tan 2\theta$ or similar. Using the double angle formula for tangent: $$\tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta}$$ However, there's no direct identity relating $\cot 4\theta$ to $\tan 2\theta$ as in the options. 6. Therefore, the correct expression for $\frac{dy}{dx}$ is: $$\frac{dy}{dx} = -b \cot 4\theta$$ 7. None of the options A, B, or C correctly match this derivative. **Final answer:** D. None of these