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Question

12.-111-ry = sin

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Solution

The given equation is y= sin 1 ( 1 x 2 1+ x 2 )

y= sin 1 ( 1 x 2 1+ x 2 )(1)

Let x=tanθ θ= tan 1 x

Hence, y= sin 1 ( 1 tan 2 θ 1+ tan 2 θ ) (2)

Also, cos( 2θ )= 1 tan 2 θ 1+ tan 2 θ (3)

From (2) and (3),

y= sin 1 ( 1 tan 2 θ 1+ tan 2 θ ) = sin 1 ( cos( 2θ ) ) = sin 1 ( sin( π 2 2θ ) ) = π 2 2θ

So, y= π 2 2 tan 1 x

Differentiate both sides,

dy dx =0 2 1+ x 2

Thus, the derivative of sin 1 ( 1 x 2 1+ x 2 ) is 2 1+ x 2 .


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