Question

Find the second order derivatives of $\log \left(\log x\right)$

Answer 1

Let $y=\log \left(\log x\right)$
$\Rightarrow \frac{dy}{dx}=\frac{d}{dx}\left[\log \right(\log x\left)\right]=\frac{1}{\log x}.\frac{d}{dx}\left(\log x\right)=\frac{1}{x\log x}=(x\log x{)}^{-1}$
$\therefore$ $\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left[\right(x\log x{)}^{-1}]=(-1).(x\log x{)}^{-2}.\frac{d}{dx}\left(x\log x\right)$
$=\frac{-1}{(x\log x{)}^2}.[\log x.\frac{d}{dx}(x)+x\frac{d}{dx}(\log x\left)\right]$
$=\frac{-1}{(x\log x{)}^2}[\log x.1+x\frac{1}{x}]=\frac{-(1+\log x)}{(x\log x{)}^2}$