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#### Provide Solution For  R.D.Sharma Maths Class 12 Chapter 10 Differentiation Exercise 10.3  Question 44 Maths Textbook Solution.

Answer:$\frac{dy}{dx}=\frac{-6}{\sqrt{1-4x^{2}}}$

Hint:

\begin{aligned} &\frac{d}{d x}\left(x^{n}\right)=n x^{n-1} \\ &\frac{d}{d x}(\text { Constant })=0 \end{aligned}

Given:

Solution:

\begin{aligned} &y=\cos ^{-1}(2 x)+2 \cos ^{-1} \sqrt{1-4 x^{2}}, \\ &-\frac{1}{2}

Put$2x=\cos \theta$

So

$y=\cos ^{-1}(\operatorname{coc} \theta)+2 \cos ^{-1} \sqrt{1-\cos ^{2} \theta}$

Using,$\sin ^{2}\theta +\cos ^{2}\theta =1$

\begin{aligned} &y=\cos ^{-1}(\cos \theta)+2 \cos \sqrt{\sin ^{2} \theta} \\ &y=\cos ^{-1}(\cos \theta)+2 \cos ^{-}(\sin \theta) \\ &y=\cos ^{-1}(\cos \theta)+2 \cos \left(\cos ^{-1}\left(\frac{\pi}{2}-\theta\right)\right)-(i) \end{aligned}

Considering limit,

\begin{aligned} &-\frac{1}{2}<\mathrm{x}<0 \\ &-1<2 \mathrm{x}<0 \\ &-1-\cos \theta<0 \\ &\frac{\mathrm{\pi}}{2}<\theta<\mathrm{\pi} \end{aligned}

And,

\begin{aligned} &-\frac{\mathrm{\pi}}{2}>-\theta>-\mathrm{\pi} \\ &\left(\frac{\pi}{2}-\frac{\pi}{2}\right)>\left(\frac{\pi}{2}-\theta\right)>\left(\frac{\mathrm{\pi}}{2}-\pi\right) \\ &0>\left(\frac{\pi}{2}-\theta\right)>-\frac{\pi}{2} \end{aligned}

So, from equation (i)

$\mathrm{y}=\theta+2\left[-\left(\frac{\pi}{2}-\theta\right)\right]$

$\left\{\operatorname{Since}, \cos ^{-1}(\cos \theta)=0 \text { if } \theta \varepsilon[0, \pi], \cos ^{-1} \cos (\theta)=-\theta, \text { if } \theta \varepsilon[-\pi \cdot 0]\right\}$

\begin{aligned} &\mathrm{y}=\theta-2 \times \frac{\pi}{2}+2 \theta \\ &\mathrm{y}=-\pi+3 \theta \\ &\mathrm{y}=-\pi+3 \cos ^{-1}(2 \mathrm{x}) \end{aligned}

Differentiating its with respect to $x$ using chain rule

$\frac{d y}{d x}=0+3\left[\frac{-1}{\sqrt{1-(2 x)^{2}}}\right] \frac{d}{d x}(2 x)$

As we Know,

\begin{aligned} &\frac{\mathrm{d}}{\mathrm{dx}}(\text { constant })=0 \\ &\frac{\mathrm{d}}{\mathrm{dx}}\left(\cos ^{-1} \mathrm{x}\right)=\frac{-1}{\sqrt{1-\mathrm{x}^{2}}} \\ &\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-3}{\sqrt{1-4 \mathrm{x}^{2}}}-(2) \\ &\frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{6}{\sqrt{1-4 \mathrm{x}^{2}}} \end{aligned}