`Let y=e^(2x)dotT h e n((d^2y)/(dx^2))((d^2x)/(dy^2))i s`

Options

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

Answer

C

Explanation

Given , ` y = e^(2x ) rArr (dy)/(dx) = 2a ^(2x) rArr (d^(2) y)/(dx^(2)) = 4a^(2x)`
Again consider ` y= e^(2x) rArr log y = 2x `
`rArr x= (1)/(2) ""log y`
`(dx)/(dy) = (1)/(2y) rArr (d^(2) x)/(dy^(2)) = - (1)/(2y^(2)) = - (1)/(2) e^(-4x)`
` therefore ((d^(2) y)/(ax^(2))) ((d^(2)x)/(dy^(2))) = 4e^(2x) (-(1)/(2)e^(-4x))`
` = - 2 e^(-2x)`.