Question

Find $\frac{d^{2}y}{d{x}^2}$ for $y=\log \left(\log x\right)$

Answer 1

$\begin{array}{cc}y& =\log \left(\log x\right)\\ \frac{dy}{dx}& =\frac{1}{logx}\cdot \frac{1}{x}\text{By chain Rule.}\\ & =\frac{1}{x\log x}\\ & =(x\log x{)}^{-1}\end{array}$

$\begin{array}{cc}\frac{d^{2}y}{d{x}^2}& =-1(x\log x{)}^{-2}\cdot (\log x+x\frac{1}{x})\\ & =\frac{-1}{(x\log x{)}^2}(1+\log x)\end{array}$