Derivative of $ \log _{2} (\log_{3}(\log_{5}b)) $

Question

I am supposed to find the derivative of $f(b)= \log _{2} (\log_{3}(\log_{5}b)) $ How would you calculate it? I know rules for derivation of logarithms but I don't know how to apply it in this case. Thanks

Answer 1

$f(x)=\log_2(\log_3(\log_5(x)))$, find $f'(x)|_{x=b}$

Chain rule: $(f(g(x)))'=f'(g(x))g'(x)$

and

$(\log_a x)'=\left(\cfrac {\ln x}{\ln a}\right)'=\cfrac 1{x\ln a }$

Apply above rule:

\begin{align} f'(x)&=\cfrac 1{\ln 2 \log_3(\log_5(x))}\cdot (\log_3(\log_5(x)))'\\ &=\cfrac 1{\ln 2 \ln 3 \log_3(\log_5(x))\log_5(x)}\cdot (\log_5(x))'\\ &=\cfrac 1{\ln 2 \cdot \ln 3 \cdot \ln 5\log_3(\log_5(x))\cdot \log_5(x) \cdot x} \end{align}

Replace $x$ with $b$ to get $f'(b)$