Step 1: Understanding the Concept:
Since the sequence of square roots is infinite, we can replace the recurring part of the expression with the variable \( y \) itself. Step 2: Detailed Explanation:
Given: \( y = \sqrt{\sin x + \sqrt{\sin x + \dots}} \).
We can write this as:
\[ y = \sqrt{\sin x + y} \]
Squaring both sides:
\[ y^{2} = \sin x + y \]
Differentiating both sides with respect to \( x \):
\[ \frac{d}{dx}(y^{2}) = \frac{d}{dx}(\sin x) + \frac{d}{dx}(y) \]
\[ 2y \frac{dy}{dx} = \cos x + \frac{dy}{dx} \]
Rearranging terms with \( \frac{dy}{dx} \):
\[ 2y \frac{dy}{dx} - \frac{dy}{dx} = \cos x \]
\[ \frac{dy}{dx} (2y - 1) = \cos x \]
\[ \frac{dy}{dx} = \frac{\cos x}{2y - 1} \]. Step 3: Final Answer:
The derivative is \( \frac{\cos x}{2y - 1} \).