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Solution for If Y = (Tan−1 X)2, Then Prove that (1 + X2)2 Y2 + 2x(1 + X2)Y1 = 2 ? - CBSE (Commerce) Class 12 - Mathematics

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Question

If y = (tan−1 x)2, then prove that (1 + x2)2 y2 + 2x(1 + x2)y1 = 2 ?

Solution

Here,

\[y = \left( \tan^{- 1} x \right)^2 \]

\[\text { Differentiating w . r . t . x, we get }\]

\[ y_1 = \frac{2 \tan^{- 1} x}{1 + x^2}\]

\[\text { Differentiating again w . r . t . x, we get }\]

\[ y_2 = \frac{2 - 4x \tan^{- 1} x}{\left( 1 + x^2 \right)^2}\]

\[ \Rightarrow y_2 = \frac{2}{\left( 1 + x^2 \right)^2} - \frac{2 \tan^{- 1} x \times 2x}{\left( 1 + x^2 \right)^2}\]

\[ \Rightarrow y_2 = \frac{2}{\left( 1 + x^2 \right)^2} - \frac{2x y_1}{\left( 1 + x^2 \right)}\]

\[ \Rightarrow \left( 1 + x^2 \right)^2 y_2 = 2 - 2x\left( 1 + x^2 \right) y_1 \]

\[ \Rightarrow \left( 1 + x^2 \right)^2 y_2 + 2x\left( 1 + x^2 \right) y_1 = 2\]

Hence proved.

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Solution for question: If Y = (Tan−1 X)2, Then Prove that (1 + X2)2 Y2 + 2x(1 + X2)Y1 = 2 ? concept: Simple Problems on Applications of Derivatives. For the courses CBSE (Commerce), CBSE (Arts), PUC Karnataka Science, CBSE (Science)
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