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

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

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

Solution:

Here it is given that,

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

This can be written as:

$\inline y=\sqrt{\cos x+y}$

Squaring on both sides, we get:

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

Differentiating (1) w.r.t x,

\inline \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}

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

Hence, it is proved.