Question

If e^y (x + 1) = 1, show that `(d^2y)/(dx^2) = (dy/dx)^2`.

Answer 1

Given, e^y (x + 1) = 1  ....(1)

Differentiating (1) w.r.t. x, we get

`d/dx e^y (x + 1) = d/dx (1)`

`e^y d/dx (x + 1) + (x + 1) d/dx e^y = 0`

`e^y(1) + (x + 1) e^y dy/dx = 0`

`e^y + (x + 1) e^y dy/dx = 0`

`(x + 1)e^y dy/dx = -e^y`

`(x + 1) dy/dx = -1`

`dy/dx = (-1)/(x + 1)`

Now square both sides:

`(dy/dx)^2 = 1/(x + 1)^2`

Differentiating both sides again with respect to x,

`(d^2 y)/dx^2 = - d/dx 1/(x + 1)`

= `- d/dx (x + 1)^-1`

= `-(-1) (x + 1)^(-1-1)`

= 1 (x + 1)⁻²

`(d^2 y)/dx^2 = 1/(x + 1)^2`

Hence, `(d^2 y)/dx^2 = (dy/dx)^2`