Question

Find the second order derivative of the function.

log (log x)

Answer 1

Let, y = log (log x)

Differentiating both sides with respect to x,

`dy/dx = d/dx log (log x)`

= `1/(log x). d/dx (log x)`

= `1/(log x) xx 1/x`

= `1/(x log x)`

= (x log x)⁻¹

Differentiating both sides again with respect to x,

`d/dx [dy/dx] = d/dx [(x log x)^-1]`

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

= `-1 (x log x)^-2 [x d/dx (log x) + log x d/dx (x)]`

= `-1 xx 1/(x log x)^2 [x xx 1/x + log x (1)]`

= `(-(1 + log x))/ (x log x)^2`