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"If "y=(tan x)^((tan x)^(tan x))," then ...

`"If "y=(tan x)^((tan x)^(tan x))," then find "(dy)/(dx).`

Text Solution

Verified by Experts

The correct Answer is:
`y log y sec^(2)x[log tan x + 1 +(1)/(tan x log tan x)]`
Taking logarithm on both sides, we get
`log y = (tan x)^(tan x)log tan x`
`therefore" "log log y = [ tan x log tan x ] + log log tan x `
Differentiating w.r.t x, we get
`(1)/(y log y )(dy)/(dx)=sec^(2) x log tan x + tan x (sec^(2)x)/(tan x)+(1)/(log tan x )xx(sec^(2)x)/(tan x)`
`therefore" "(dy)/(dx)=y log y sec^(2) x [ log tan x + 1+ //(tan x log tan x )]`
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