Question

Find the second order derivatives of each of the following functions :
$\log \left(\log x\right)$

Answer 1

Let $y=log\left(logx\right).$ Then,
$\frac{dy}{dx}=\frac{1}{logx}\frac{d}{dx}\left(logx\right)=\frac{1}{xlogx}=(xlogx{)}^{-1}$
$\Rightarrow \frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left\{\right(xlogx{)}^{-1}\}$
$\Rightarrow \frac{d^{2}y}{d{x}^2}=-\left(xlogx{)}^{-2}\frac{d}{dx}\right(xlogx)=-\frac{1}{(xlogx{)}^2}(1+logx)=\frac{-(1+logx)}{(xlogx{)}^2}$