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  • Please solve RD Sharma class 12 chapter Differentiation exercise 10.2 question 75 maths textbook solution

Please solve RD Sharma class 12 chapter Differentiation exercise 10.2 question 75 maths textbook solution

Answers (1)

Answer: \frac{2}{3}

Hint: you must know the rules of solving derivatives of trigonometric function

Given: f(x)=\sqrt{\frac{\sec x-1}{\sec x+1}}


Find : f^{\prime}\left(\frac{\pi}{3}\right)
 

Solution:

f(x)=\sqrt{\frac{\sec x-1}{\sec x+1}}

        =\sqrt{\frac{1-\cos x}{1+\cos x}} \quad\left[\therefore \sec x=\frac{1}{\cos x}\right]

 

Now rationalize

        =\sqrt{\frac{1-\cos x}{1+\cos x} \times \frac{1-\cos x}{1-\cos x}}

f(x)=\frac{1-\cos x}{\sin x}

         =\frac{1}{\sin x}-\frac{\cos x}{\sin x}

f(x)=\operatorname{cosec} x-\cot x

 

Differentiate with respect to x,

\begin{aligned} &f^{\prime}(x)=-\operatorname{cosec} x \cot x-\left(-\operatorname{cosec}^{2} x\right)\\ \\ &f^{\prime}\left(\frac{\pi}{3}\right)=-\operatorname{cosec}\left(\frac{\pi}{3}\right) \cot \left(\frac{\pi}{3}\right)+\operatorname{cosec}^{2}\left(\frac{\pi}{3}\right) \end{aligned}

\begin{aligned} &=\frac{-2}{\sqrt{3}} \times \frac{1}{\sqrt{3}}+\left(\frac{2}{\sqrt{3}}\right)^{2} \\\\ &=\frac{-2}{3}+\frac{4}{3} \\\\ &\Rightarrow \frac{-2+4}{3} \Rightarrow \frac{2}{3} \end{aligned}

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