Find the derivative y=(tan(x))^(6/x)

Jaylene Hunter

Jaylene Hunter

Answered question

2022-07-20

Find the derivative
y = ( tan ( x ) ) 6 / x

Answer & Explanation

akademiks1989rz

akademiks1989rz

Beginner2022-07-21Added 16 answers

Given, y = ( tan ( x ) ) 6 / x
Taking log on both sides, we get
log y = log [ tan ( x ) ] 6 / x log y = 6 x log ( tan x )     ( log a x = x log a ) log y = 6 log ( tan ( x ) ) x
Now, differentiating both sides w.r.t "x"
d d x ( log y ) = d d x [ 6 log ( tan ( x ) ) x ] 1 y d y d x = 6 d d x log ( tan ( x ) ) x 1 y d y d x = 6 { x d d x log ( tan x ) log ( tan x ) d x d x x 2 }       ( d d x ( u v ) = v d d x u u d d x v v 2 ) = 6 { x 1 tan x d d x ( tan x ) log ( tan x ) ( 1 ) x 2 } = 6 { x tan x sec 2 x log ( tan x ) x 2 }
1 y = 6 { x sec 2 x tan x log ( tan x ) x 2 tan x } d d x = 6 y { x sec 2 x tan x log ( tan x ) x 2 tan x }
Put the value of y = ( tan x ) 6 / x we get
d y d x = 6 ( tan x ) 6 / x x 2 tan ( x ) { x sec 2 x tan x log ( tan x ) } = 6 x 2 ( tan x ) 6 x 1 { x sec 2 x tan x log ( tan x ) } d y d x = 6 x 2 ( tan ( x ) ) 6 x x { x sec 2 x tan x log ( tan x ) }

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