Determine    dy /dx for the following and simplify where possible: y = (secx+3 lntanx)1/4no

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to differentiate the function with respect to x, using the chain rule, such that:

`(dy)/(dx) = (d(sec x + ln tan x)^(1/4))/(dx)`

`(dy)/(dx) = (1/4)(sec x + ln tan x)^(1/4 - 1)*(sec x + ln tan x)'`

`(dy)/(dx) = (1/4)(sec x + ln tan x)^(-3/4)(sec x*tan x + (sec^2 x)/(tan x))`

`(dy)/(dx) = (1/4)(1/(root(4)((sec x + ln tan x)^3)))(sec x*tan x + (sec^2 x)/(tan x))`

`(dy)/(dx) = (1/(4*tan x*root(4)((sec x + ln tan x)^3)))(sec x*tan^2 x + sec^2 x)`

`(dy)/(dx) = (sec x(tan^2 x + sec x))/(4*tan x*root(4)((sec x + ln tan x)^3))`

`(dy)/(dx) = ((1/cos x)(tan^2 x + sec x))/(4*(sin x/cos x)*root(4)((sec x + ln tan x)^3))`

`(dy)/(dx) = (tan^2 x + sec x)/(4sin x*root(4)((sec x + ln tan x)^3))`

Hence, differentiating the given function with respect to x yields `(dy)/(dx) = (tan^2 x + sec x)/(4sin x*root(4)((sec x + ln tan x)^3)).`

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