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  • Please solve RD Sharma class 12 chapter Higher Order Derivatives exercise 11.1 question 41 maths textbook solution

Please solve RD Sharma class 12 chapter Higher Order Derivatives exercise 11.1 question 41 maths textbook solution

Answers (1)

Answer:

Proved

Hint:

You must know the derivative of cot inverse x

Given:

y=\left(\cot ^{-1} x\right)^{2}, \text { prove } y_{2}\left(x^{2}+1\right)^{2}+2 x\left(x^{2}+1\right) y_{1}=2

Solution:

y=\left(\cot ^{-1} x\right)^{2}

        \begin{aligned} &\frac{d y}{d x}=2 \cot ^{-1} x\left(\frac{-1}{1+x^{2}}\right)\\ &\left(1+x^{2}\right) \frac{d y}{d x}=-2 \cot ^{-1} x\\ &\text { Again differentiating }\\ &\left(1+x^{2}\right)^{2} \frac{d^{2} y}{d x^{2}}+2 x \frac{d y}{d x}=-2\left(\frac{-1}{1+x^{2}}\right) \end{aligned}

        \begin{aligned} &\left(1+x^{2}\right)^{2} \frac{d^{2} y}{d x^{2}}+2 x \frac{d y}{d x}=\left(\frac{2}{1+x^{2}}\right) \\ &\left(1+x^{2}\right)^{2} \frac{d^{2} y}{d x^{2}}+2 x\left(1+x^{2}\right) \frac{d y}{d x}=2 \\ &y_{2}\left(x^{2}+1\right)^{2}+2 x\left(x^{2}+1\right) y_{1}=2 \end{aligned}

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Gurleen Kaur

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