If
\(y = tan \: x^{tan \: x^{tan \: x}}\)
, then dy/dx is

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)

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