Question

If , show that

Answer 1

Let, e y ( x+1 )=1.

Simplify the equation as,

e y ( x+1 )=1 e y = 1 1+x (1)

Taking log on both sides, we get,

y=log 1 ( x+1 )

The first order derivative is obtained by differentiating the function with respect to x.

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

Again differentiate the above function with respect to x.

d 2 y d x 2 = d dx { −1 x+1 } =−{ −1 ( 1+x ) 2 } = 1 ( 1+x ) 2

Further simplify.

d 2 y d x 2 = ( −1 1+x ) 2 =( dy dx )

Hence, it is proved that d 2 y d x 2 = ( dy dx ) 2 .