# Differentiate the following w.r.t. $x :$$log ( log \: x ), x > 1 \begin{array}{1 1} \large \frac{1}{x \log x} \\ \large \frac{x}{ \log x} \\ \large \frac{\log x}{x } \\x \log x \end {array} ## 1 Answer Toolbox: • According to the Chain Rule for differentiation, given two functions f(x) and g(x), and y=f(g(x)) \rightarrow y' = f'(g(x)).g'(x). • \; \large \frac{d(\log x)}{dx}$$=\large\frac{1}{x}$
Given $y = \log (\log x)$
This is of the form $y = f(g(x)$, where $g(x) = \log x$, so we can apply the chain rule of differentiation.
$\; \large \frac{d(\log x)}{dx}$$=\large\frac{1}{x}$
$\Rightarrow g'(x) = \large\frac{1}{x}$
According to the Chain Rule for differentiation, given two functions $f(x)$ and $g(x)$, and $y=f(g(x)) \rightarrow y' = f'(g(x)).g'(x)$.
$\Rightarrow y' = \large \frac{1}{\log x}\;\frac{1}{x} = \frac{1}{x \log x}$