Question

find the derivative of y=ln[ln[ln[lnx]

Answer 1

lol

Answer 2

repeated applications of the chain rule

Answer 3

start with
\[\frac{1}{\ln(\ln(\ln(x)))}\] then take the derivative of \(\ln(ln(ln(x)))\)

Answer 4

Sometimes, it helps to use substitutions to see the function differently and approach it systematically:\[y=\ln(\ln(\ln(\ln x)))\]
Let \(f=\ln x\):
\[y=\ln(\ln(\ln f))\]
Let \(g=\ln f\):
\[y=\ln(\ln g)\]
Let \(h=\ln g\):
\[y=\ln h\]
Keeping in mind that \(f,g,h\) are functions of \(x\), apply the chain rule:
\[y'=\frac{h'}{h}\]
Subbing back, you have
\[\large y'=\frac{(\ln g)'}{\ln g}\\
\large y'=\frac{\frac{g'}{g}}{\ln(\ln f)}\\
\large y'=\frac{\frac{(\ln f)'}{\ln f}}{\ln(\ln f)}\\
\large y'=\frac{\frac{(\ln x)'}{\ln (\ln x)}}{\ln(\ln (\ln x))}\\
\large y'=\frac{\frac{\frac{1}{x}}{\ln (\ln x)}}{\ln(\ln (\ln x))}\\
y'=\cdots \]