Question

The derivative of

\[ y = \sqrt{\sin x + \sqrt{\sin x + \sqrt{\sin x + \cdots \infty}}} \] 

is ______.

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

Hint:

For $y=\sqrt{f(x)+\sqrt{f(x)+\dots}}$, the derivative is always \[ \dfrac{f'(x)}{2y-1} \]

Answer 1

The Correct Option is D

Step 1: Simplify the expression
The given function can be written as $y=\sqrt{\sin x+y}$ since the nested part is also $y$.

Step 2: Square both sides
$y^2=\sin x+y$

Step 3: Differentiate with respect to $x$
$2y\dfrac{dy}{dx}=\cos x+\dfrac{dy}{dx}$

Step 4: Rearrange to find $\dfrac{dy{dx}$}
$2y\dfrac{dy}{dx}-\dfrac{dy}{dx}=\cos x$ $\dfrac{dy}{dx}(2y-1)=\cos x$ $\dfrac{dy}{dx}=\dfrac{\cos x}{2y-1}$
Final Answer: (D)