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  • Provide Solution For R.D.Sharma Maths Class 12 Chapter 10 Differentiation Exercise 10.3 Question 42 Maths Textbook Solution.

Provide Solution For  R.D.Sharma Maths Class 12 Chapter 10 Differentiation Exercise 10.3  Question 42 Maths Textbook Solution.

Answers (1)

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

Hint:

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

Given:

y=\cos ^{-1}(2 x)+2 \cos ^{-1} \sqrt{1+4 x^{2}}, 0<x<\frac{1}{2}

Solution:

Put 2x=\cos \theta

So,

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

\begin{aligned} &0<x=1 \\ &0<2 x<1 \\ &0<\cos \theta=1 \\ &0<\theta<\frac{\pi}{2} \\ &\text { And } 0>-\theta>-\frac{\pi}{2} \\ &\frac{n}{2}>\left(\frac{n}{2}-\theta\right)>0 \end{aligned}

So From eq (i)

y=\theta +2\left ( \frac{\pi}{2}-\theta \right )                                                                                \left \{ Since \cos ^{-1}\left ( \cos \theta \right )=0\: if\: \theta \varepsilon \left [ 0,n \right ] \right \}

\begin{aligned} &y \Rightarrow \theta+\pi-2 \theta \\ &y \Rightarrow \pi-C \\ &y=\pi-\cos ^{-1}(2 x) \\ &\text { Since, } 2 x=\cos \theta \end{aligned}

Differentiating it with respect to x

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

Since,\frac{d}{dx}\left ( constant \right )=0;

\begin{aligned} &\frac{d}{d x}\left(\cos ^{-1} x\right)=\frac{1}{\sqrt{1-x^{2}}} \\ &\frac{d y}{d x}=\frac{1}{\sqrt{1-4 x^{2}}}(2) \\ &\frac{d y}{d x}=\frac{2}{\sqrt{1-4 x^{2}}} \end{aligned}



 

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