PUC Karnataka Science Class 12Department of Pre-University Education, Karnataka
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# Solution for If Y = (Tan−1 X)2, Then Prove that (1 + X2)2 Y2 + 2x(1 + X2)Y1 = 2 ? - PUC Karnataka Science Class 12 - Mathematics

ConceptSimple Problems on Applications of Derivatives

#### 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 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.
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