Step 1: Understanding the Concept:
The given function is an infinite nested radical. The key to solving such problems is to recognize the repeating pattern and express the function in a simple implicit form, which can then be differentiated.
Step 2: Key Formula or Approach:
Since the expression is infinite, we can write:
\[ y = \sqrt{\sin x + y} \]
By squaring both sides, we get an implicit equation relating x and y. We then use implicit differentiation to find $\frac{dy}{dx}$.
Step 3: Detailed Explanation:
The given function is $y = \sqrt{\sin x + \sqrt{\sin x + \dots}}$.
We can rewrite this as:
\[ y = \sqrt{\sin x + y} \]
Square both sides to remove the outer square root:
\[ y^2 = \sin x + y \]
Rearrange the terms to prepare for differentiation:
\[ y^2 - y = \sin x \]
Now, differentiate both sides with respect to x using implicit differentiation.
\[ \frac{d}{dx}(y^2 - y) = \frac{d}{dx}(\sin x) \]
\[ 2y\frac{dy}{dx} - 1\frac{dy}{dx} = \cos x \]
Factor out $\frac{dy}{dx}$ on the left side:
\[ (2y - 1)\frac{dy}{dx} = \cos x \]
Solve for $\frac{dy}{dx}$:
\[ \frac{dy}{dx} = \frac{\cos x}{2y - 1} \]
Now, we check the options. The denominators are in the form $1-2y$. Let's rewrite our result to match this form.
\[ 2y - 1 = -(1 - 2y) \]
So,
\[ \frac{dy}{dx} = \frac{\cos x}{-(1 - 2y)} = \frac{-\cos x}{1 - 2y} \]
Step 4: Final Answer:
The derivative $\frac{dy}{dx}$ is $\frac{-\cos x}{1 - 2y}$. Therefore, option (D) is correct.