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Find the second order derivatives of the functions given in \(\log ( \log x) \)

$\begin{array}{1 1} (x\log x)^{-2} \\-\Large\frac{1+\log x}{(x\log x)^2} \\(x\log x)^{-1} \\\Large\frac{1+\log x}{(x\log x)^2} \end{array} $

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  • $y=f(x)$
  • $\large\frac{dy}{dx}$$=f'(x)$
  • $\large\frac{d^2y}{dx^2}=\frac{d}{dx}\big(\frac{dy}{dx}\big)$
  • $\large\frac{d}{dx}$$(\log x)=\large\frac{1}{x}$
Step 1:
$y=\log(\log x)$
Differentiating with respect to $x$
$\large\frac{dy}{dx}=\frac{1}{\log x}\frac{d}{dx}$$(\log x)$
$\quad\;=\large\frac{1}{x\log x}$
$\quad\;=(x\log x)^{-1}$
Step 2:
$\large\frac{d^2y}{dx^2}=\frac{-1}{(x\log x)^2}$$(x.\large\frac{1}{x}$$+\log x.1)$
$\large\frac{d^2y}{dx^2}=\frac{-1}{(x\log x)^2}$$(1+\log x)$
$\quad\;=-\large\frac{1+\log x}{(x\log x)^2}$
answered May 13, 2013 by sreemathi.v
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