Solution:
Given
\(\begin{array}{l}y = tan \: x^{tan \: x^{tan \: x}}\end{array} \)
Taking log
log y = tan xtan x log (tan x)
Taking log
log (log y) = log [tan xtan x log (tan x)]
= tan x log tan x + log (log tan x)
Differentiate w.r.t.x
(1/log y)× (1/y)×dy/dx = (tan x/tan x) sec2 x + log (tan x ) sec2x + 1/(log tan x) × (1/tan x)× sec2x
(dy/dx) × (1/y log y) = sec2x + log (tan x) sec2x + sec2x/tan x log tan x
= sec2x (1 + log (tan x) + 1/tan x log tan x)
dy/dx = y log y sec2x (1 + log (tan x) + 1/tan x log tan x)
