Question

If , for, −1 < x <1, prove that

Answer 1

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( y−x ) ( x+y )( x−y )=xy( y−x )

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+x−x ( 1+x ) 2 dy dx =− 1 ( 1+x ) 2

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