Derivative of y={ x+{ x+{ x+ }}} { is }

Correct Answer is : cos⁡x2y−1 x/2 y-12y−1cosx

Correct Answer is : cos⁡x2y−1 x/2 y-12y−1cosx

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MHTCET 5 May 2025 Shift 2 Paper

Question : 111 of 150

Marks: +1, -0

Derivative of

y=\sqrt{\sin x+\sqrt{\sin x+\sqrt{\sin x+\ldots \ldots \ldots \ldots \ldots \ldots}}} \text { is }

Solution:

Concept:

For an infinite nested radical of the form

y=\sqrt{f(x)+y}

, differentiate after squaring.

Explanation:

Given

y=\sqrt{\sin x+\sqrt{\sin x+\sqrt{\sin x+\cdots}}}

Since the pattern repeats infinitely, we rewrite it as:

y=\sqrt{\sin x+y}

Squaring both sides:

y^2=\sin x+y

Differentiate both sides with respect to

x

2y\frac{dy}{dx}=\cos x+\frac{dy}{dx}

Rearrange terms:

(2y-1)\frac{dy}{dx}=\cos x

Hence:

\frac{dy}{dx}=\frac{\cos x}{2y-1}

Answer:

\frac{dy}{dx}=\frac{\cos x}{2y-1}

So the correct option is D.

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