Explanation

(a)
$f(x) = \log_{5}{(\log_{7}{x)}}$
$\Rightarrow f(x) = \log_{5}\left( \frac{\log_{e}x}{\log_{e}7} \right)$
$\Rightarrow f(x) = \log_{5}{\log_{e}{x - \log_{5}{\log_{e}7}}}$
$\Rightarrow f(x) = \frac{\log_{e}{\log_{e}x}}{\log_{e}5} - \log_{5}{\log_{e}7}$
On differentiating w.r.t. $x$, we get
$f^{'}(x) = \frac{1}{x\log_{e}{xlog_{e\ 5\ }}} - 0$
$= \frac{1}{\frac{x\log_{e}x}{\log_{e}7}\log_{e}{7.\log_{e}5}}$
$= \frac{1}{x\log_{7}{x\log{7.\log 5}}}$