$\log \left[ {\log \left( {\log {x^5}} \right)} \right]$

Let $y = \log \left[ {\log \left( {\log {x^5}} \right)} \right]$
therefore,$\frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {\log \left( {\log \log {x^5}} \right)} \right]$

$= \frac{1}{{\log \log {x^5}}} \cdot \frac{d}{{dx}}\left( {\log \cdot \log {x^5}} \right)$

$= \frac{1}{{\log \log {x^5}}} \cdot \left( {\frac{1}{{\log {x^5}}}} \right) \cdot \frac{d}{{dx}}\log {x^5}$

$= \frac{1}{{\log \log {x^5}}} \cdot \frac{1}{{\log {x^5}}} \cdot \frac{d}{{dx}}(5\log x) = \frac{5}{{x \cdot \log \left( {\log {x^5}} \right) \cdot \log \left( {{x^5}} \right)}}$

$\Rightarrow \frac{dy}{dx} = \frac{5}{{x \cdot \log \left( {\log {x^5}} \right) \cdot \log \left( {{x^5}} \right)}}$