Differentiate w.r.t x, \[\log \left( {\log x} \right)\]

Hint:
Use the chain rule to differentiate the given function since the function is a composite function and chain rule tells us how to differentiate a composite function which is given as
\[f\left( {g\left( x \right)} \right) = f'\left( {g\left( x \right)} \right).g'\left( x \right)\]
Here we need to differentiate the given function with respect to x where differentiation of expression is the rate of change of a function with respect to independent variables.

Derivation of a function is basically a measure of sensitivity to change of function value with change in the argument where argument refers to the input whose output is to be found. Derivatives are useful in finding the slope of an equation, maxima, and minima of a function when the slope is zero and is also used to check a function, whether it is increasing or decreasing.

Complete step by step solution:
Let \[y = \log (\log x) - (i)\]
Since the given function is a composite function, so we use the chain rule to differentiate equation which is given as
\[f\left( {g\left( x \right)} \right) = f'\left( {g\left( x \right)} \right).g'\left( x \right)\]
Hence by using the chain rule in equation (i), we can write
\[
  \dfrac{{d\left( y \right)}}{{dx}} = \dfrac{{d\left( {\log \left( {\log x} \right)} \right)}}{{dx}} \\
  \dfrac{{dy}}{{dx}} = \dfrac{1}{{\log x}} \times \dfrac{{d\left( {\log x} \right)}}{{dx}} \\
  \dfrac{{dy}}{{dx}} = \dfrac{1}{{\log x}} \times \dfrac{1}{x} \\
   = \dfrac{1}{{x\log x}} \\
 \][Since\[\dfrac{d}{{dx}}\left( {\log x} \right) = \dfrac{1}{x}\]]

Hence the differentiate of \[\log \left( {\log x} \right)\]with respect to x \[ = \dfrac{1}{{x\log x}}\]

The important formula used:
\[\dfrac{d}{{dx}}\left( {\log x} \right) = \dfrac{1}{x}\]

Note:
As a constant term does not contain any variables with them when they are differentiated, then their value is zero. Derivation of a function is represented in\[\dfrac{a}{b}\], where \[a\] is the function which is being differentiated and b its independent variable by which function is being differentiated written as \[\dfrac{{dy}}{{dx}}\] where y is the function.