If `y=x+e^x` , find `(d^2x)/(dy^2)` .

Options

A. `(1)/((1+e^(x))^(2))`
B. `-(e^(x))/((1+e^(x))^(2))`
C. `-(e^(x))/((1+e^(x))^(3))`
D. `e^(x)`

Answer

C

Explanation

We have,
`y=x+e^(x)`
`implies" "(dy)/(dx)=1+e^(x)`
`implies" "(dx)/(dy)=(1)/(1+e^(2))`
`implies" "(d^(2)x)/(dy^(2))=(d)/(dy)((1)/(1+e^(x)))`
`implies" "(d^(2)x)/(dy^(2))=-(1)/((1+e^(x))^(2))(d)/(dy)(1+e^(x))`
`implies" "(d^(2)x)/(dy^(2))=-(1)/((1+e^(x))^(2))e^(x)(dx)/(dy)=(-e^(x))/((1+e^(x))^(3))`