Question

If \(y=\sqrt{\sin x + \sqrt{\sin x + \cdots}}\), then \(\frac{dy}{dx}\) is

• A. \(\frac{\sin x}{2y-1}\)
• B. \(\frac{\cos x}{1-2y}\)
• C. \(\frac{\cos x}{2y-1}\)
• D. 0

Hint:

For infinite radicals, replace the repeating part by the variable itself.

Answer 1

The Correct Option is C

Concept: Infinite nested expression $\Rightarrow$ set \(y\) equal to whole.

Step 1: Form equation.
\[ y=\sqrt{\sin x + y} \]

Step 2: Square.
\[ y^2 = \sin x + y \]

Step 3: Differentiate.
\[ 2y\frac{dy}{dx} = \cos x + \frac{dy}{dx} \]

Step 4: Solve.
\[ (2y-1)\frac{dy}{dx} = \cos x \] \[ \Rightarrow \frac{dy}{dx} = \frac{\cos x}{2y-1} \]