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  • Please solve RD Sharma class 12 chapter Higher Order Derivatives exercise 11.1 question 33 maths textbook solution

Please solve RD Sharma class 12 chapter Higher Order Derivatives exercise 11.1 question 33 maths textbook solution

Answers (1)

Answer:

-cot\: y.cosec^{2}y

Hint:

You must know the derivative of cos inverse function

Given:

y=\cos ^{-1} x, \text { find } \frac{d^{2} y}{d x^{2}} \text { in terms of } y \text { alone }

Solution:

y=\cos ^{-1} x

        \begin{aligned} &\text { Differentiate the equation w.r.t } x\\ &\frac{d y}{d x}=\frac{d\left(\cos ^{-1} x\right)}{d x}\\ &=\frac{-1}{\sqrt{1-x^{2}}}=-\left(1-x^{2}\right)^{\frac{-1}{2}}\\ &\frac{d^{2} y}{d x^{2}}=\frac{d\left[-\left(1-x^{2}\right)^{\frac{-1}{2}}\right]}{d x} \end{aligned}

        \begin{aligned} &=-\left(\frac{-1}{2}\right)\left(1-x^{2}\right)^{\frac{-3}{2}} \times(-2 x) \\ &=\frac{1}{2 \sqrt{\left(1-x^{2}\right)^{3}}} \times(-2 x) \\ &\frac{d^{2} y}{d x^{2}}=\frac{-x}{\sqrt{\left(1-x^{2}\right)^{3}}} \end{aligned}

        \begin{aligned} &y=\cos ^{-1} x\\ &x=\cos y\\ &\text { Put in above equation }\\ &\frac{d^{2} y}{d x^{2}}=\frac{-\cos y}{\sqrt{\left(1-\cos ^{2} y\right)^{3}}} \end{aligned}

        \begin{aligned} &\frac{d^{2} y}{d x^{2}}=\frac{-\cos y}{\sqrt{\sin ^{3} y}} \\ &=\frac{-\cos y}{\sin ^{3} y} \\ &=\frac{-\cos y}{\sin y} \times \frac{1}{\sin ^{2} y} \\ &\therefore \frac{d^{2} y}{d x^{2}}=-\cot y \cdot \operatorname{cosec}^{2} y \end{aligned}

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Gurleen Kaur

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