Question

Find the second order derivative of the given functions.

log (log x).

Answer 1

Ley y = log (log x)

Differentiating twicely w.r.t. x, we get

$\frac{dy}{dx}=\frac{1}{logx}\frac{d}{dx}\left(logx\right)=\frac{1}{logx}.\frac{1}{x}=\frac{1}{xlogx}and\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(\frac{1}{xlogx}\right)=\frac{d}{dx}(xlogx{)}^{-1}=-1(xlogx{)}^{-2}\frac{d}{dx}\left(xlogx\right)=-\frac{1}{(xlogx{)}^2}[x\frac{d}{dx}logx+logx\frac{d}{dx}(x\left)\right]\frac{-1}{(xlogx{)}^2}(x.\frac{1}{x}+logx.1)=\frac{-(1+logx)}{(xlogx{)}^2}$