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Question

If , for, −1 < x <1, prove that

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Solution

Let, x 1+y +y 1+x =0.

Simplify the given equation.

x 1+y +y 1+x =0 x 1+y =y 1+x

By squaring both sides of the above equation, we get,

x 2 ( 1+y )= y 2 ( 1+x ) x 2 + x 2 y= y 2 +x y 2 x 2 y 2 =xy( yx ) ( x+y )( xy )=xy( yx )

Further simplify the above equation.

( x+y )=xy x+y+xy=0 y( 1+x )=x y= x ( 1+x )

Differentiate both sides with respect to x.

dy dx = d dx ( x 1+x ) ={ ( 1+x ) d dx ( x )x d dx ( 1+x ) ( 1+x ) 2 } = 1+xx ( 1+x ) 2 dy dx = 1 ( 1+x ) 2

Hence, it is proved that dy dx = 1 ( 1+x ) 2 .


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