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Provide solution RD Sharma maths class 12 chapter 10 differentiation exercise 10.6 question 2 maths textbook solution

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Answer: \frac{d y}{d x}=\frac{\sin x}{1-2 y}

Hint: The value of y is given as infinite series. If a term is deleted from an infinite series, it remains the same in this case.

Given: y=\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+\ldots \ldots \ldots . .+\infty}}}


Here it is given that,

              y=\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+\ldots \ldots \ldots . .+\infty}}}

This can be written as:

              y=\sqrt{\cos x+y}

Squaring on both sides, we get:

           y^{2}=\left ( \cos x+y \right )                                                                                                   …(1)

Differentiating (1) w.r.t x,

                          \begin{aligned} &2 y \frac{d y}{d x}=-\sin x+\frac{d y}{d x} \\ &\frac{d y}{d x}(2 y-1)=-\sin x \\ &\frac{d y}{d x}=\frac{-\sin x}{2 y-1} \end{aligned}

                          \therefore \frac{d y}{d x}=\frac{-\sin x}{2 y-1}

Hence, it is proved.

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